1. Formula
Geometric formula
► Perimeter of rectangle = (length + width) × 2
C = (a + b) × 2
► Area of rectangle = length × width
S = ab
► Perimeter of square = sides Length×4
C=4a
►The area of the square =side length×side length
S=a.a=a
►The area of the triangle =base×height÷2
S=ah÷2
►The sum of the interior angles of the triangle =180 degrees
►The parallelogram Area = base The radius of the circle = diameter ÷ 2 (r = d ÷ 2)
► The circumference of the circle = pi × diameter = pi × radius × 2
C = πd = 2πr
► The area of the circle = pi × radius × radius
S = πr × r
► The volume of the cuboid = length × width × height
V = abh
► The volume of the cube = edge length × edge length × edge length V = aaa
► The side area of the cylinder : The side area of the cylinder is equal to the perimeter of the base multiplied by the height
S=ch=πdh=2πrh
►The surface area of the cylinder : The surface area of the cylinder is equal to the circumference of the base multiplied by the height plus the area of the circles at both ends
S=ch+2s=ch+2πr×r
►The volume of the cylinder : The volume of the cylinder The volume is equal to the base area times the height
V=Sh
►The volume of the cone =1/3 base Meter = 10 centimeters
1 centimeters = 10 millimeters
►1 square meter = 100 square decimeters
1 square decimeter = 100 square centimeters
1 square centimeters = 100 square millimeters
►1 cubic meter = 1000 cubic decimeters
1 cubic decimeter = 1000 Cubic centimeter
1 cubic centimeter = 1000 cubic millimeters
►1 ton = 1000 kilograms
1 kilogram = 1000 grams = 1 kilogram = 2 pounds
►1 hectare = 10,000 square meters
1 mu = 666.666 square meters
► 1 liter = 1 cubic centimeter Meter = 1000 ml
1 ml = 1 cubic centimeter
1 yuan = 10 jiao
1 jiao = 10 cents
1 yuan = 100 cents
1 century = 100 years
1 year = 12 months
big months (31 days): 18 months
Small months (30 days) are: 49 months
February 28 days in ordinary years, February 29 days in leap years
365 days in an ordinary year, 366 days in a leap year
1 day = 24 hours
1 hour = 60 minutes = 3600 seconds
1 Minutes = 60 seconds
Quantity relationship
► Number of copies per copy × number of copies = total number
total number ÷ number of copies = number of copies
total number of copies ÷ number of copies = number per copy
►1 multiple × multiple = how many multiples
how many multiples ÷ 1 multiple = multiple
multiples ÷ multiples = 1 multiple
► Speed × time = distance
distance ÷ speed = time
distance ÷ time = speed
►Unit price × quantity = total price
total price ÷ unit price = quantity
total price ÷ Quantity = unit price
► Work efficiency × working time = total amount of work
total amount of work ÷ work efficiency = working time
total amount of work ÷ working time = work efficiency
► addend + addend = sum
and - an addend = another addend
►minuend minuend-minuend=difference
minuend-difference=minuend
difference+minuend=minuend
► factor ×factor=product
product÷one factor= Another factor
►Divisor ÷Divisor=Quotient
Dividend ÷Quotient=Divisor
Quotient × Divisor=Divisor
Special Problems
►Meeting Problem
Meeting Distance=Speed and ×Meeting Time
Meeting Time=Meeting Distance ÷Speed and
Speed sum=distance to meet ÷time to meet
►Catching problem
Catching distance =speed difference
Downstream speed = Still water speed + Water flow speed
Countercurrent speed = Still water speed - Water flow speed
Still water speed = (Downstream speed + Countercurrent speed) ÷2
Water flow speed = (Downstream speed - Countercurrent speed) ÷2
(2) The formula for two ships sailing in opposite directions:
Ship A's speed along the water + Ship B's speed against the water = Ship A's still water speed + Ship B's still water speed
(3) The formula for two ships sailing in the same direction:
Ship's still water speed behind (front) - front ( After) The speed of the ship in still water = the distance between the two ships decreases (increases) speed
► Concentration problem
The weight of the solute + the weight of the solvent = the weight of the solution
The weight of the solute ÷ the weight of the solution × 100% = concentration
The weight of the solution × concentration = weight of solute
weight of solute ÷ concentration = weight of solution
► Profit and discount issues
profit = selling price - cost
profit margin = profit ÷ cost × 100% = (selling price ÷ cost - 1) × 100 %
Increase or decrease amount = principal × increase or decrease percentage
Discount = actual selling price ÷ original selling price × 100% (discount <1)>
Interest = principal × interest rate × time
After-tax interest = principal × interest rate × time × (1-5%)
► Engineering problem
Work efficiency × working time = total amount of work
Total amount of work ÷ work time = work efficiency
Total amount of work ÷ work efficiency = working time
1 ÷ working time = completion per unit time What fraction of the total amount of work
1 ÷ what fraction can be completed per unit time = working time
2. Numbers and number operations
concepts
► Integers
1, the meaning of integers
Natural numbers and 0 are both integers.
. Natural numbers
When we count objects, the 1, 2, 3... used to represent the number of objects are called natural numbers. There is no object in
, which is represented by 0. 0 is also a natural number.
. Counting units
One (one), ten, one hundred, one thousand, ten thousand, one hundred thousand, one million, ten million, billion... are all counting units. Where "one" is the basic unit of counting. 0 1's are 10, 10 10's are 100... The progress rate between each two adjacent counting units is 10. This counting method is called decimal notation.
, digits
The counting units are arranged in a certain order, and the positions they occupy are called digits.
. How to read integers: from high to low, read level by level. When reading "100 million" or "10,000", read it according to the pronunciation of "one" level first, and then add the word "billion" or "ten thousand" at the end. The 0 at the end of each level is not read out, and if there are several consecutive 0s in other digits, only one zero is read.
. How to write integers: from high to low, write level by level. If there is no unit on any digit, write 0 on that digit.
. A large multi-digit number. For the convenience of reading and writing, it is often rewritten as a number using "ten thousand" or "hundred million" as the unit. Sometimes you can omit the number after a certain digit of the number and write it as an approximate number as needed.
⑴ Accurate number: In real life, for the convenience of counting, a larger number can be rewritten into a number in units of tens of thousands or billions. The rewritten number is the exact number of the original number. For example, if 1254300000 is rewritten as a number in tens of thousands, it is 1254.3 million; if it is rewritten as a number in hundreds of millions, it is 1.2543 billion.
⑵ Approximate number: According to actual needs, we can also omit the mantissa after a certain digit of a larger number and use an approximate number to represent it. For example: 1302490015 omitting the last digit after billion is 1.3 billion. ⑶ Rounding method: To find an approximate number, look at the number in the highest digit of the mantissa. If it is smaller than 5, round it off. If it is 5 or greater, round it off and advance the mantissa by 1. This method of finding approximate numbers is called rounding.
. Comparison of integer sizes: The number with more digits is larger. If the digits are the same, look at the highest digit. If the number at the highest digit is larger, the number is larger. If the number at the highest digit is the same, look at the next digit. , whichever digit has a larger number will be larger. And so on.
► Decimals
1, the meaning of decimals
Divide the integer 1 evenly into 10 parts, 100 parts, 1000 parts... The resulting tenths, hundredths, thousandths... can be expressed as decimals. For example, 1/10 is recorded as 0.1, and 7/100 is recorded as 0.07.
One decimal place represents tenths, two decimal places represent hundredths, and three decimal places represent thousandths...
A decimal is composed of an integer part, a decimal part and a decimal point part. The dot in a number is called the decimal point, the number to the left of the decimal point is called the integer part, the number to the left of the decimal point is called the integer part, and the number to the right of the decimal point is called the decimal part.
The first digit to the right of the decimal point is called the tenth place, and the counting unit is one tenth (0.1); the second digit is called the hundredth, and the counting unit is one hundredth (0.01)... The largest counting unit of the decimal part is tenths. One, there is no smallest unit of counting. The number of digits in the decimal part is called the number of decimal places. For example, 0.36 is two decimal places, and 3.066 is three decimal places.
In decimals, the rate between each two adjacent counting units is 10. The rate of progress between the highest fractional unit "one-tenth" of the decimal part and the lowest unit "one" of the whole number part is also 10.
. How to read decimals: When reading decimals, the integer part is read as an integer, the decimal point is read as "dot", and the decimal part reads the numbers on each digit in sequence from left to right.
. How to write decimals: When writing decimals, write the integer part as an integer. The decimal point is written in the lower right corner of the ones place. The decimal part writes the numbers on each digit in sequence.
. Compare the sizes of decimals: first look at their integer parts, the number with the larger integer part is larger; if the integer parts are the same, the number with the larger tenth place is larger; the numbers in the tenth place are also the same, The number with a larger percentile is larger...
. Classification of decimals
⑴ Pure decimal: A decimal whose integer part is zero is called a pure decimal. For example: 0.25 and 0.368 are both pure decimals.
⑵ With decimals: Decimals whose integer part is not zero are called with decimals. For example: 3.25 and 5.26 are both with decimals.
⑶ Finite decimal: The number of digits in the decimal part is a finite decimal, which is called a finite decimal. For example: 41.7, 25.3, 0.23 are all finite decimals.
⑷ Infinite decimals: The digits in the decimal part are infinite decimals, which are called infinite decimals. For example: 4.33... 3.1415926...
⑸ Infinite non-repeating decimal: The decimal part of a number, the number arrangement is irregular and the number of digits is infinite. Such a decimal is called an infinite non-repeating decimal. For example: π
⑹ Repeating decimal: The decimal part of a number has one or several numbers that repeatedly appear in sequence. This number is called a recurring decimal. For example: 3.555... 0.0333... 12.109109...
The decimal part of a recurring decimal, and the numbers that repeatedly appear in sequence are called the cyclic section of this recurring decimal. For example: the cyclic section of 3.99... is "9", and the cyclic section of 0.5454... is "54".
⑺ Pure recurring decimal: The recurring section starts from the first digit of the decimal part, which is called a pure recurring decimal. For example: 3.111... 0.5656...
⑻ Mixed cyclic decimal: The cyclic section does not start from the first digit of the decimal part, it is called a mixed cyclic decimal. 3.1222... 0.03333...
When writing recurring decimals, for simplicity, you only need to write a recurring section for the recurring part of the decimal, and put a dot on the first and last digits of this recurring section. If the loop section has only one number, just click a dot on it.
► Fraction
1. The meaning of fraction
Divide the unit "1" evenly into several parts, and the number representing such one or several parts is called a fraction.
In a fraction, the horizontal line in the middle is called the fraction line; the number below the fraction line is called the denominator, which indicates how many parts the unit "1" is divided into equally; the number below the fraction line is called the numerator, which indicates how many parts there are.
divides the unit "1" evenly into several parts, indicating the number of one part, which is called a fractional unit.
. How to read fractions: When reading fractions, read the denominator first, then "divided" and then read the numerator. The numerator and denominator are read as integers.
. How to write fractions: first write the fraction line, then the denominator, and finally the numerator. Write it as an integer.
. Comparing the sizes of fractions:
⑴ For fractions with the same denominators, the fraction with the larger numerator is larger.
⑵ For fractions with the same numerator, the fraction with the smaller denominator is larger.
⑶ For fractions with different denominators and numerators, the common denominator is usually converted into a fraction with a common denominator, and then the magnitude is compared.
⑷ If the fractions being compared are mixed numbers, first compare their integer parts, and the mixed number with the larger integer part will be larger; if the integer parts are the same, then compare their fraction parts, and the mixed number with the larger fraction part will be larger.
. Classification of fractions
can be divided into: true fractions, improper fractions , and mixed fractions
according to the different conditions of the numerator, denominator and integer part. ⑴ Proper fractions: The fraction with a smaller numerator than the denominator is called a true fraction. The true score is less than 1.
⑵ Improper fraction: A fraction whose numerator is greater than the denominator or whose numerator and denominator are equal is called an improper fraction. An improper fraction is greater than or equal to 1.
⑶ Mixed numbers: Improper fractions can be written as numbers composed of integers and proper fractions, usually called mixed numbers.
. The relationship between fractions and division and the basic properties of fractions
⑴ Division is an operation with operation symbols; fraction is a kind of number. Therefore, it should generally be stated that the dividend is equivalent to the numerator, but it cannot be said that the dividend is the numerator.
⑵ Since fractions and division are closely related, the basic properties of fractions can be derived based on the property of "the quotient remains unchanged" in division.
⑶ The numerator and denominator of a fraction are multiplied or divided by the same number (except 0), and the size of the fraction remains unchanged. This is called the basic property of fractions. It is the basis for to reduce and common fractions.
. Reduction and general division
⑴ The fraction whose numerator and denominator are coprime numbers is called the simplest fraction.
⑵ Converting a fraction into a fraction that is equal to it but has a smaller numerator and denominator is called a reduction.
⑶ Method of reduction: use the common denominator (except 1) of the numerator and denominator to remove the numerator and denominator; usually divide until the simplest fraction is obtained.
⑷ Convert fractions with different denominators into fractions with the same denominator that are equal to the original fractions, which are called common denominators.
⑸ Method of common fraction: first find the lowest common multiple of and of the original denominators, and then convert each fraction into a fraction using this lowest common multiple as the denominator.
. Reciprocal
⑴ Two numbers whose product is 1 are reciprocals of each other.
⑵ To find the reciprocal of a number (except 0), just swap the positions of the numerator and denominator of the number.
⑶ The reciprocal of 1 is 1, 0 has no reciprocal
► Percentage
1. The meaning of percentage
represents the percentage of one number to another number. It is called a percentage, also called a percentage or a percentage. Percentages are usually expressed with "%". The percent sign is a symbol that represents a percentage.
. How to read percentages: When reading percentages, read the percent first, then read the number before the percent sign. When reading, read it as an integer.
. How to write percentages: Percents are usually not written as fractions, but are represented by adding a percent sign "%" after the original numerator.
. Interchange between percentages, discounts, and percentages:
For example: 30% off is 30%, 25% off is 75%, percentage is a few tenths, if one percentage is 10%, then 65% is 65%.
. Taxation and interest:
Tax rate: the ratio of tax payable to various incomes.
interest rate: the percentage of interest to principal. Calculated annually or monthly as specified by the bank.
Interest calculation formula: Interest = principal × interest rate × time
. The difference between percentages and fractions mainly has the following three points:
⑴ The meanings are different. A percentage is "a number that expresses what percent of one number is another number." It can only express the multiple relationship between two numbers, but cannot express a specific quantity. For example: It can be said that 1 meter is 20% of 5 meters, but it cannot be said that "the length of a piece of rope is 20% of meters." Therefore, the unit name cannot be followed by the percentage. A fraction is "a number that divides the unit '1' evenly into several parts and represents such one or several parts." Fractions can not only express the multiple relationship between two numbers, such as: A's number is 3, B's number is 4, A's number is B's number?; it can also express a certain quantity, such as: 犌Э歭米, etc.
⑵ The application scope is different. Percents are often used in surveys, statistics, analysis and comparisons in production, work and life. Fractions are often used in measurements and calculations when integer results are not available.
⑶ The writing form is different. Percents are usually not written as fractions, but are expressed using the percent sign "%". For example: forty-five percent, written as: 45%; the denominator of the percentage is fixed at 100, therefore, no matter how many common divisors there are between the numerator and the denominator of the percentage, the fraction is not reduced; the numerator of the percentage can be a natural number, or Can be decimal. The numerator of a fraction can only be a natural number, and its representation forms are: true fraction, improper fraction, and mixed number. If the calculation result is not the simplest fraction, it must be divided into the simplest fraction through reduction, and if it is an improper fraction, it must be turned into a mixed number.
. Mutual conversion of numbers
⑴ Convert decimals into fractions: How many decimals there are in the original, just write a few zeros after 1 as the denominator, remove the decimal point from the original decimal as the numerator, and the fraction can be reduced.
⑵ Convert a fraction to a decimal: Use the denominator to remove the numerator. Those that can be divided into finite decimals are converted into finite decimals. Some cannot be divided into finite decimals, and those that cannot be converted into finite decimals are generally kept to three decimal places.
⑶ A simplest fraction. If the denominator contains no other prime factors except 2 and 5, the fraction can be converted into a finite decimal; if the denominator contains prime factors other than 2 and 5, the fraction cannot be converted into a finite decimal. decimal.
⑷ Convert decimals into percentages: Just move the decimal point two places to the right and add a percent sign at the end.
⑸ Convert percentages to decimals: To convert percentages to decimals, just remove the percent sign and move the decimal point two places to the left.
⑹ Convert fractions into percentages: Usually first convert the fraction into a decimal (when division is not possible, usually keep three decimal places), and then convert the decimal into a percentage.
⑺ Convert a percentage into a decimal: First rewrite the percentage into a fraction, and then reduce the percentage to the simplest fraction.
► Divisibility of numbers
1, the meaning of integer division
Integer a is divided by integer b (b ≠ 0), the quotient of the division is an integer without remainder , we say that a can be divided by b, or b can Divide a.
The meaning of divisibility: When number A is divided by number B and the quotient is an integer or a finite decimal and the remainder is 0, we say that number A can be divided by number B (or that number B can divide number A) The numbers A and B here can be natural numbers or decimals (number B cannot be 0).
2, divisors and multiples
⑴ If the number a can be divided by the number b (b ≠ 0), a is called a multiple of b, and b is called the divisor of a (or factor of a). Multiples and divisors are interdependent.
⑵ The number of divisors of a number is limited, the smallest divisor is 1, and the largest divisor is itself.
⑶ The number of multiples of a number is infinite. The smallest multiple is itself, and there is no largest multiple.
, odd numbers and even numbers
⑴ Natural numbers can be divided into odd numbers and even numbers according to the characteristics of whether they can be divided by 2.
① A number that is divisible by 2 is called an even number. 0 is also an even number.
② A number that is not divisible by 2 is called an odd number.
⑵ Operational properties of odd and even numbers:
① The sum of two adjacent natural numbers is an odd number, and the product is an even number.
② Odd number + odd number = even number, odd number + even number = odd number, even number + even number = even number; odd number - odd number = even number,
odd number - even number = odd number, even number - odd number = odd number, even number - even number = even number; odd number × odd number = odd number, Odd number × even number = even number, even number × even number = even number.
. Characteristics of divisibility
⑴ Numbers with units digits of 0, 2, 4, 6, and 8 can all be divisible by 2.
⑵ Any number whose units digit is 0 or 5 can be divisible by 5.
⑶ If the sum of the ones digits of a number is divisible by 3, then the number can be divisible by 3.
⑷ If the sum of a number's digits is divisible by 9, then the number can be divisible by 9.
⑸ A number divisible by 3 may not necessarily be divisible by 9, but a number divisible by 9 must be divisible by 3.
⑹ If the last two digits of a number are divisible by 4 (or 25), then the number can be divisible by 4 (or 25).
⑺ If the last three digits of a number are divisible by 8 (or 125), then this number can be divisible by 8 (or 125).
, prime numbers and composite numbers
⑴ If a number has only two divisors, 1 and itself, such a number is called a prime number (or prime number). The prime numbers within 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
⑵ If a number has other divisors besides 1 and itself, such a number is called a composite number. For example, 4, 6, 8, 9, and 12 are all composite numbers.
⑶ 1 is neither a prime number nor a composite number. Except for 1, all natural numbers are either prime numbers or composite numbers. If natural numbers are classified according to the number of their divisors, they can be divided into prime numbers, composite numbers and 1.
. Decomposing prime factors
⑴ Prime factors
Each composite number can be written as the multiplication of several prime numbers. Each prime number is a factor of this composite number and is called a prime factor of this composite number. For example, 15=3×5, 3 and 5 are called prime factors of 15.
⑵ Decomposing prime factors
expresses a composite number in the form of multiplying prime factors, which is called decomposing prime factors. Short division is often used to factor prime factors. First divide by prime numbers that can divide the composite number, keep dividing until the quotient is a prime number, and then write the divisor and quotient in the form of continuous multiplication.
⑶ Common factor (approximate) number
The common factors of several numbers are called the common factors of these numbers. The largest one is called the greatest common factor of these numbers.
Two numbers whose common factors are only 1 are called coprime numbers. Two numbers that are in a mutually prime relationship have the following situations: ① Relatively prime with any natural number;
② Two adjacent natural numbers are relatively prime;
③ When the composite number is not a multiple of a prime number, the composite number is relatively prime with the prime number ;
④ When the common divisor of two composite numbers is only 1, the two composite numbers are relatively prime. If any two of several numbers are relatively prime, it is said that these numbers are mutually prime.
If the smaller number is a divisor of the larger number, then the smaller number is the greatest common divisor of the two numbers.
If two numbers are relatively prime, their greatest common divisor is 1.
⑷ Common multiples
① The common multiples of several numbers are called the common multiples of these numbers. The largest one is called the greatest common multiple of these numbers.
The method to find the greatest common divisor of several numbers is: first use the common divisors of these numbers to divide continuously until the obtained quotient only has a common divisor of 1, and then multiply all the divisors together to find the product. This product is The greatest common divisor of these numbers.
② The common multiples of several numbers are called the common multiples of these numbers, and the smallest one among them is called the least common multiple of these numbers.
The method for finding the least common multiple of several numbers is: first divide by the common divisors of these numbers (or part of them) until they are relatively prime (or pairwise relatively prime), and then sum up all the divisors Multiply the quotients together to find the product. This product is the least common multiple of these numbers.
If the larger number is a multiple of the smaller number, then the larger number is the least common multiple of the two numbers.
If two numbers are relatively prime, then the product of the two numbers is their least common multiple.
The number of common divisors of several numbers is limited, but the number of common multiples of several numbers is infinite.
Properties and laws
(1) The law of constant quotient
The law of constant quotient: In division, the dividend and divisor are expanded or reduced by the same times at the same time, and the quotient remains unchanged.
(2) Properties of decimals
Properties of decimals: Add zero or remove zero at the end of the decimal, the size of the decimal remains unchanged.
(3) The movement of the decimal point position causes changes in the size of the decimal . Move the decimal point one place to the right, and the original number will be expanded by 10 times; move the decimal point two places to the right, and the original number will be expanded by 100 times; move the decimal point three places to the right. digit, the original number will be expanded 1000 times... . Move the decimal point one place to the left, and the original number will be reduced 10 times; move the decimal point two places to the left, and the original number will be reduced 100 times; move the decimal point three places to the left, The original number is reduced by 1000 times... . When the decimal point is moved to the left or to the right and there are not enough digits, "0" must be used to make up the digits.
(4) Basic properties of fractions
Basic properties of fractions: The numerator and denominator of a fraction are multiplied or divided by the same number (except zero), and the size of the fraction remains unchanged.
(5) The relationship between fractions and division . Dividend ÷ divisor = dividend/divisor . Because zero cannot be used as a divisor, the denominator of a fraction cannot be zero. . The dividend is equivalent to the numerator, and the divisor is equivalent to the denominator.
Operation rules
(1) The rules of the four arithmetic operations of integers . Integer addition:
The operation of combining two numbers into one number is called addition.
In addition, the number added is called the addend, and the number added is called the sum. The addend is the part number and the sum is the total number.
addend + addend = sum one addend = sum - another addend
. Integer subtraction:
It is known that the sum of two addends and one of the addends, the operation of finding the other addend is called subtraction.
In subtraction, the known sum is called the minuend, the known addend is called the subtrahend, and the unknown addend is called the difference. The minuend is the total number, and the subtrahend and difference are the partial numbers respectively.
addition and subtraction are inverse operations of each other.
3, integer multiplication :
The simple operation of finding the sum of several identical addends is called multiplication.
In multiplication, the same addends and the number of the same addends are called factors.The sum of the same addends is called the product.
In multiplication, 0 multiplied by any number gets 0. Multiplying 1 with any number gets any number.
One factor
In division, the known product is called the dividend, a known factor is called the divisor, and the required factor is called the quotient.
multiplication and division are the inverse operations of each other.
In division, 0 cannot be used as the divisor. Since 0 multiplied by any number is 0, any number divided by 0 cannot get a definite quotient.
dividend ÷ divisor = quotient divisor = dividend ÷ quotient dividend = quotient × divisor
. Power:
The operation of finding the product of several identical factors is called exponentiation. For example, 3 × 3 =32
(2) Four decimal arithmetic operations
1, decimal addition:
The meaning of decimal addition is the same as that of integer addition. It is an operation that combines two numbers into one number.
. Decimal subtraction:
The meaning of decimal subtraction is the same as that of integer subtraction. Given the sum of two addends and one of the addends, find the operation of the other addend.
. Multiplication of decimals:
The meaning of multiplying decimals by integers is the same as the meaning of multiplication of integers, which is a simple operation to find the sum of several identical addends; the meaning of multiplying a number by a pure decimal is to find the tenths and hundredths of the number. , a few thousandths...how much is it.
. Decimal division:
The meaning of decimal division is the same as that of integer division. It is the operation of finding the product of two factors and one of the factors to find the other factor.
(3) The four arithmetic operations of fractions
1. Fraction addition:
The meaning of fraction addition is the same as that of integer addition. It is an operation that combines two numbers into one number. . Subtraction of fractions: The meaning of
subtraction of fractions is the same as that of subtraction of integers. Given the sum of two addends and one of the addends, find the operation of the other addend. . Fractional multiplication:
The meaning of fractional multiplication is the same as that of integer multiplication, which is a simple operation to find the sum of several identical addends. . Fractional division:
The meaning of fractional division is the same as that of integer division. It is the operation of finding the product of two factors and one of the factors to find the other factor.
(4) Laws of operation
1, Laws of addition
⑴ Commutative law of addition:
When two numbers are added, the positions of the addends are exchanged, and their sum remains unchanged, that is, a+b=b+a.
⑵ Associative law of addition:
To add three numbers, add the first two numbers first, and then add the third number; or add the last two numbers first, and then add their sum to the first number. unchanged, that is (a+b)+c=a+(b+c).
. The law of multiplication
⑴ The commutative law of multiplication:
When two numbers are multiplied, their product remains unchanged at the position of the exchange factor, that is, a×b=b×a.
⑵ Associative law of multiplication:
To multiply three numbers, first multiply the first two numbers and then multiply by the third number; or first multiply the last two numbers and then multiply by the first number. Their The product remains unchanged, that is, (a×b)×c=a×(b×c).
⑶Distributive law of multiplication:
The sum of two numbers is multiplied by a number. You can multiply the two addends by this number respectively, and then add the two products, that is (a+b)×c=a×c +b×c.
⑷ Extension of the distributive law of multiplication:
The difference between two numbers is multiplied by one number. You can first multiply them by this number and then subtract them, that is (a-b) ×c=a×c-b×c
. The law of subtraction
⑴ To subtract several numbers from a number continuously, you can subtract the sum of all subtrahends from this number without changing the difference, that is, a-b-c=a-(b+c).
⑵ To subtract two numbers from one number continuously, you can subtract the second subtrahend first, and then subtract the first subtrahend, that is, a-b-c=a-c-b.
. The law of division operation
⑴ If a number is divided by two numbers continuously, it can be divided by the set of these two numbers, that is, a÷b÷c=a÷(b×c).
⑵ If a number is divided by two numbers continuously, it can be divided by the second divisor first, and then divided by the first divisor, that is, a÷b÷c=a÷c÷b.
, other
a-b+c=a+c-b
a-b+c=a+(b-c)
a÷b×c=a×c÷b
a÷b×c=a÷(b÷c)
. Change rule: In multiplication, one factor remains unchanged, the other factor expands (or shrinks) several times, and the product also expands (or shrinks) by the same multiple.
promotion: One factor is expanded by A times, the other factor is expanded by B times, and the product is expanded by AB times.
One factor reduces it by A times, the other factor reduces it by B times, and the product is reduced by AB times.
. Invariant quotient property: In division, the dividend and divisor are expanded (or reduced) by the same multiple at the same time, and the quotient remains unchanged. m≠0 a÷b=(a×m) ÷(b×m)=(a÷m) ÷(b÷m)
Generalization: The dividend is expanded (or reduced) A times, the divisor remains unchanged, and the quotient also expands ( or shrink) A times.
The dividend remains unchanged, the divisor expands (or shrinks) by A times, but the quotient shrinks (or expands) by A times.
can make some calculations simple by using the changing laws of products and the invariant properties of quotients. But be careful about the remainder in division with a remainder. For example: 8500÷200= You can divide by reducing the dividend and divisor by 100 times at the same time, that is, 85÷2=, the quotient remains unchanged, but the remainder 1 at this time is reduced by 100 times, so the original remainder should be 100.
(5) Calculation method
1, integer addition calculation rule:
The same digits are aligned, starting from the low digit, and whichever digit adds up to ten, advance one to the previous digit.
. Calculation rules for integer subtraction:
Align the same digits and add them up from the low digit. If the number in any digit is not enough to subtract, go back one digit from the previous digit and make ten, merge it with the number in the original digit, and then subtract.
. Calculation rules for integer multiplication:
First use the number in each digit of one factor to multiply the number in each digit of another factor. Use the number in each digit of the factor to multiply, and the ends of the multiplied numbers will be aligned. One digit, and then add up the multiplied numbers.
. Calculation rules for integer division:
Divide from the high digit of the dividend first. How many digits the divisor is depends on the first few digits of the dividend. If it is not enough to divide, look at one more digit. Which digit of the dividend is divided, the quotient is Which one is written above. If there is not enough quotient of 1 in any bit, "0" should be added to occupy the place. The remainder of each division must be less than the divisor.
. Decimal multiplication rules:
First calculate the product according to the calculation rules of integer multiplication, then look at how many decimals there are in the factors, count them from the right side of the product, and click the decimal point; if there are not enough digits, use "0" "Complement."
. Calculation rules for decimal division when the divisor is an integer:
First divide according to the rules of integer division. The decimal point of the quotient should be aligned with the decimal point of the dividend; if there is still a remainder at the end of the dividend, add "0" after the remainder, and then Continue to remove.
. Calculation rules for division when the divisor is a decimal:
first moves the decimal point of the divisor to make it an integer, and also moves the decimal point of the divisor a few places to the right (if there are not enough digits, add "0"), and then follow the division method when the divisor is an integer. Calculate according to the rules.
. Calculation method of adding and subtracting fractions with the same denominator:
To add and subtract fractions with the same denominator, only add and subtract the numerators, leaving the denominator unchanged.
. Calculation method of adding and subtracting fractions with different denominators:
first use common denominators, and then calculate according to the rules of adding and subtracting fractions with the same denominator.
0. Calculation method of adding and subtracting mixed fractions:
Add and subtract the integer parts and fractional parts respectively, and then combine the resulting numbers.
1. Calculation rules for fraction multiplication:
When multiplying a fraction by an integer, use the product of the numerator of the fraction and the integer as the numerator, and the denominator remains unchanged; when multiplying a fraction by a fraction, use the product of the numerators as the numerator, and the product of the denominators as the denominator. .
2. Calculation rules for fraction division:
Number A divided by number B (except 0) is equal to the reciprocal of number A times number B.
(6) The order of operations
1, the order of operations of the four arithmetic operations on decimals is the same as the order of the four arithmetic operations on integers.
2. The order of operations of the four arithmetic operations on fractions is the same as the order of the four arithmetic operations on integers. . Mixed operations without parentheses: same-level operations are performed from left to right; two-level operations are performed first, multiplication and division, and then addition and subtraction.. Mixed operations with parentheses: first calculate the items inside the parentheses, then calculate the items inside the square brackets, and finally calculate the items outside the brackets. , first-level operations: addition and subtraction are called first-level operations. . Second-level operations: Multiplication and division are called second-level operations.
. Natural numbers
When we count objects, the 1, 2, 3... used to represent the number of objects are called natural numbers. There is no object in
, which is represented by 0. 0 is also a natural number.
. Counting units
One (one), ten, one hundred, one thousand, ten thousand, one hundred thousand, one million, ten million, billion... are all counting units. Where "one" is the basic unit of counting. 0 1's are 10, 10 10's are 100... The progress rate between each two adjacent counting units is 10. This counting method is called decimal notation.
, digits
The counting units are arranged in a certain order, and the positions they occupy are called digits.
. How to read integers: from high to low, read level by level. When reading "100 million" or "10,000", read it according to the pronunciation of "one" level first, and then add the word "billion" or "ten thousand" at the end. The 0 at the end of each level is not read out, and if there are several consecutive 0s in other digits, only one zero is read.
. How to write integers: from high to low, write level by level. If there is no unit on any digit, write 0 on that digit.
. A large multi-digit number. For the convenience of reading and writing, it is often rewritten as a number using "ten thousand" or "hundred million" as the unit. Sometimes you can omit the number after a certain digit of the number and write it as an approximate number as needed.
⑴ Accurate number: In real life, for the convenience of counting, a larger number can be rewritten into a number in units of tens of thousands or billions. The rewritten number is the exact number of the original number. For example, if 1254300000 is rewritten as a number in tens of thousands, it is 1254.3 million; if it is rewritten as a number in hundreds of millions, it is 1.2543 billion.
⑵ Approximate number: According to actual needs, we can also omit the mantissa after a certain digit of a larger number and use an approximate number to represent it. For example: 1302490015 omitting the last digit after billion is 1.3 billion. ⑶ Rounding method: To find an approximate number, look at the number in the highest digit of the mantissa. If it is smaller than 5, round it off. If it is 5 or greater, round it off and advance the mantissa by 1. This method of finding approximate numbers is called rounding.
. Comparison of integer sizes: The number with more digits is larger. If the digits are the same, look at the highest digit. If the number at the highest digit is larger, the number is larger. If the number at the highest digit is the same, look at the next digit. , whichever digit has a larger number will be larger. And so on.
► Decimals
1, the meaning of decimals
Divide the integer 1 evenly into 10 parts, 100 parts, 1000 parts... The resulting tenths, hundredths, thousandths... can be expressed as decimals. For example, 1/10 is recorded as 0.1, and 7/100 is recorded as 0.07.
One decimal place represents tenths, two decimal places represent hundredths, and three decimal places represent thousandths...
A decimal is composed of an integer part, a decimal part and a decimal point part. The dot in a number is called the decimal point, the number to the left of the decimal point is called the integer part, the number to the left of the decimal point is called the integer part, and the number to the right of the decimal point is called the decimal part.
The first digit to the right of the decimal point is called the tenth place, and the counting unit is one tenth (0.1); the second digit is called the hundredth, and the counting unit is one hundredth (0.01)... The largest counting unit of the decimal part is tenths. One, there is no smallest unit of counting. The number of digits in the decimal part is called the number of decimal places. For example, 0.36 is two decimal places, and 3.066 is three decimal places.
In decimals, the rate between each two adjacent counting units is 10. The rate of progress between the highest fractional unit "one-tenth" of the decimal part and the lowest unit "one" of the whole number part is also 10.
. How to read decimals: When reading decimals, the integer part is read as an integer, the decimal point is read as "dot", and the decimal part reads the numbers on each digit in sequence from left to right.
. How to write decimals: When writing decimals, write the integer part as an integer. The decimal point is written in the lower right corner of the ones place. The decimal part writes the numbers on each digit in sequence.
. Compare the sizes of decimals: first look at their integer parts, the number with the larger integer part is larger; if the integer parts are the same, the number with the larger tenth place is larger; the numbers in the tenth place are also the same, The number with a larger percentile is larger...
. Classification of decimals
⑴ Pure decimal: A decimal whose integer part is zero is called a pure decimal. For example: 0.25 and 0.368 are both pure decimals.
⑵ With decimals: Decimals whose integer part is not zero are called with decimals. For example: 3.25 and 5.26 are both with decimals.
⑶ Finite decimal: The number of digits in the decimal part is a finite decimal, which is called a finite decimal. For example: 41.7, 25.3, 0.23 are all finite decimals.
⑷ Infinite decimals: The digits in the decimal part are infinite decimals, which are called infinite decimals. For example: 4.33... 3.1415926...
⑸ Infinite non-repeating decimal: The decimal part of a number, the number arrangement is irregular and the number of digits is infinite. Such a decimal is called an infinite non-repeating decimal. For example: π
⑹ Repeating decimal: The decimal part of a number has one or several numbers that repeatedly appear in sequence. This number is called a recurring decimal. For example: 3.555... 0.0333... 12.109109...
The decimal part of a recurring decimal, and the numbers that repeatedly appear in sequence are called the cyclic section of this recurring decimal. For example: the cyclic section of 3.99... is "9", and the cyclic section of 0.5454... is "54".
⑺ Pure recurring decimal: The recurring section starts from the first digit of the decimal part, which is called a pure recurring decimal. For example: 3.111... 0.5656...
⑻ Mixed cyclic decimal: The cyclic section does not start from the first digit of the decimal part, it is called a mixed cyclic decimal. 3.1222... 0.03333...
When writing recurring decimals, for simplicity, you only need to write a recurring section for the recurring part of the decimal, and put a dot on the first and last digits of this recurring section. If the loop section has only one number, just click a dot on it.
► Fraction
1. The meaning of fraction
Divide the unit "1" evenly into several parts, and the number representing such one or several parts is called a fraction.
In a fraction, the horizontal line in the middle is called the fraction line; the number below the fraction line is called the denominator, which indicates how many parts the unit "1" is divided into equally; the number below the fraction line is called the numerator, which indicates how many parts there are.
divides the unit "1" evenly into several parts, indicating the number of one part, which is called a fractional unit.
. How to read fractions: When reading fractions, read the denominator first, then "divided" and then read the numerator. The numerator and denominator are read as integers.
. How to write fractions: first write the fraction line, then the denominator, and finally the numerator. Write it as an integer.
. Comparing the sizes of fractions:
⑴ For fractions with the same denominators, the fraction with the larger numerator is larger.
⑵ For fractions with the same numerator, the fraction with the smaller denominator is larger.
⑶ For fractions with different denominators and numerators, the common denominator is usually converted into a fraction with a common denominator, and then the magnitude is compared.
⑷ If the fractions being compared are mixed numbers, first compare their integer parts, and the mixed number with the larger integer part will be larger; if the integer parts are the same, then compare their fraction parts, and the mixed number with the larger fraction part will be larger.
. Classification of fractions
can be divided into: true fractions, improper fractions , and mixed fractions
according to the different conditions of the numerator, denominator and integer part. ⑴ Proper fractions: The fraction with a smaller numerator than the denominator is called a true fraction. The true score is less than 1.
⑵ Improper fraction: A fraction whose numerator is greater than the denominator or whose numerator and denominator are equal is called an improper fraction. An improper fraction is greater than or equal to 1.
⑶ Mixed numbers: Improper fractions can be written as numbers composed of integers and proper fractions, usually called mixed numbers.
. The relationship between fractions and division and the basic properties of fractions
⑴ Division is an operation with operation symbols; fraction is a kind of number. Therefore, it should generally be stated that the dividend is equivalent to the numerator, but it cannot be said that the dividend is the numerator.
⑵ Since fractions and division are closely related, the basic properties of fractions can be derived based on the property of "the quotient remains unchanged" in division.
⑶ The numerator and denominator of a fraction are multiplied or divided by the same number (except 0), and the size of the fraction remains unchanged. This is called the basic property of fractions. It is the basis for to reduce and common fractions.
. Reduction and general division
⑴ The fraction whose numerator and denominator are coprime numbers is called the simplest fraction.
⑵ Converting a fraction into a fraction that is equal to it but has a smaller numerator and denominator is called a reduction.
⑶ Method of reduction: use the common denominator (except 1) of the numerator and denominator to remove the numerator and denominator; usually divide until the simplest fraction is obtained.
⑷ Convert fractions with different denominators into fractions with the same denominator that are equal to the original fractions, which are called common denominators.
⑸ Method of common fraction: first find the lowest common multiple of and of the original denominators, and then convert each fraction into a fraction using this lowest common multiple as the denominator.
. Reciprocal
⑴ Two numbers whose product is 1 are reciprocals of each other.
⑵ To find the reciprocal of a number (except 0), just swap the positions of the numerator and denominator of the number.
⑶ The reciprocal of 1 is 1, 0 has no reciprocal
► Percentage
1. The meaning of percentage
represents the percentage of one number to another number. It is called a percentage, also called a percentage or a percentage. Percentages are usually expressed with "%". The percent sign is a symbol that represents a percentage.
. How to read percentages: When reading percentages, read the percent first, then read the number before the percent sign. When reading, read it as an integer.
. How to write percentages: Percents are usually not written as fractions, but are represented by adding a percent sign "%" after the original numerator.
. Interchange between percentages, discounts, and percentages:
For example: 30% off is 30%, 25% off is 75%, percentage is a few tenths, if one percentage is 10%, then 65% is 65%.
. Taxation and interest:
Tax rate: the ratio of tax payable to various incomes.
interest rate: the percentage of interest to principal. Calculated annually or monthly as specified by the bank.
Interest calculation formula: Interest = principal × interest rate × time
. The difference between percentages and fractions mainly has the following three points:
⑴ The meanings are different. A percentage is "a number that expresses what percent of one number is another number." It can only express the multiple relationship between two numbers, but cannot express a specific quantity. For example: It can be said that 1 meter is 20% of 5 meters, but it cannot be said that "the length of a piece of rope is 20% of meters." Therefore, the unit name cannot be followed by the percentage. A fraction is "a number that divides the unit '1' evenly into several parts and represents such one or several parts." Fractions can not only express the multiple relationship between two numbers, such as: A's number is 3, B's number is 4, A's number is B's number?; it can also express a certain quantity, such as: 犌Э歭米, etc.
⑵ The application scope is different. Percents are often used in surveys, statistics, analysis and comparisons in production, work and life. Fractions are often used in measurements and calculations when integer results are not available.
⑶ The writing form is different. Percents are usually not written as fractions, but are expressed using the percent sign "%". For example: forty-five percent, written as: 45%; the denominator of the percentage is fixed at 100, therefore, no matter how many common divisors there are between the numerator and the denominator of the percentage, the fraction is not reduced; the numerator of the percentage can be a natural number, or Can be decimal. The numerator of a fraction can only be a natural number, and its representation forms are: true fraction, improper fraction, and mixed number. If the calculation result is not the simplest fraction, it must be divided into the simplest fraction through reduction, and if it is an improper fraction, it must be turned into a mixed number.
. Mutual conversion of numbers
⑴ Convert decimals into fractions: How many decimals there are in the original, just write a few zeros after 1 as the denominator, remove the decimal point from the original decimal as the numerator, and the fraction can be reduced.
⑵ Convert a fraction to a decimal: Use the denominator to remove the numerator. Those that can be divided into finite decimals are converted into finite decimals. Some cannot be divided into finite decimals, and those that cannot be converted into finite decimals are generally kept to three decimal places.
⑶ A simplest fraction. If the denominator contains no other prime factors except 2 and 5, the fraction can be converted into a finite decimal; if the denominator contains prime factors other than 2 and 5, the fraction cannot be converted into a finite decimal. decimal.
⑷ Convert decimals into percentages: Just move the decimal point two places to the right and add a percent sign at the end.
⑸ Convert percentages to decimals: To convert percentages to decimals, just remove the percent sign and move the decimal point two places to the left.
⑹ Convert fractions into percentages: Usually first convert the fraction into a decimal (when division is not possible, usually keep three decimal places), and then convert the decimal into a percentage.
⑺ Convert a percentage into a decimal: First rewrite the percentage into a fraction, and then reduce the percentage to the simplest fraction.
► Divisibility of numbers
1, the meaning of integer division
Integer a is divided by integer b (b ≠ 0), the quotient of the division is an integer without remainder , we say that a can be divided by b, or b can Divide a.
The meaning of divisibility: When number A is divided by number B and the quotient is an integer or a finite decimal and the remainder is 0, we say that number A can be divided by number B (or that number B can divide number A) The numbers A and B here can be natural numbers or decimals (number B cannot be 0).
2, divisors and multiples
⑴ If the number a can be divided by the number b (b ≠ 0), a is called a multiple of b, and b is called the divisor of a (or factor of a). Multiples and divisors are interdependent.
⑵ The number of divisors of a number is limited, the smallest divisor is 1, and the largest divisor is itself.
⑶ The number of multiples of a number is infinite. The smallest multiple is itself, and there is no largest multiple.
, odd numbers and even numbers
⑴ Natural numbers can be divided into odd numbers and even numbers according to the characteristics of whether they can be divided by 2.
① A number that is divisible by 2 is called an even number. 0 is also an even number.
② A number that is not divisible by 2 is called an odd number.
⑵ Operational properties of odd and even numbers:
① The sum of two adjacent natural numbers is an odd number, and the product is an even number.
② Odd number + odd number = even number, odd number + even number = odd number, even number + even number = even number; odd number - odd number = even number,
odd number - even number = odd number, even number - odd number = odd number, even number - even number = even number; odd number × odd number = odd number, Odd number × even number = even number, even number × even number = even number.
. Characteristics of divisibility
⑴ Numbers with units digits of 0, 2, 4, 6, and 8 can all be divisible by 2.
⑵ Any number whose units digit is 0 or 5 can be divisible by 5.
⑶ If the sum of the ones digits of a number is divisible by 3, then the number can be divisible by 3.
⑷ If the sum of a number's digits is divisible by 9, then the number can be divisible by 9.
⑸ A number divisible by 3 may not necessarily be divisible by 9, but a number divisible by 9 must be divisible by 3.
⑹ If the last two digits of a number are divisible by 4 (or 25), then the number can be divisible by 4 (or 25).
⑺ If the last three digits of a number are divisible by 8 (or 125), then this number can be divisible by 8 (or 125).
, prime numbers and composite numbers
⑴ If a number has only two divisors, 1 and itself, such a number is called a prime number (or prime number). The prime numbers within 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
⑵ If a number has other divisors besides 1 and itself, such a number is called a composite number. For example, 4, 6, 8, 9, and 12 are all composite numbers.
⑶ 1 is neither a prime number nor a composite number. Except for 1, all natural numbers are either prime numbers or composite numbers. If natural numbers are classified according to the number of their divisors, they can be divided into prime numbers, composite numbers and 1.
. Decomposing prime factors
⑴ Prime factors
Each composite number can be written as the multiplication of several prime numbers. Each prime number is a factor of this composite number and is called a prime factor of this composite number. For example, 15=3×5, 3 and 5 are called prime factors of 15.
⑵ Decomposing prime factors
expresses a composite number in the form of multiplying prime factors, which is called decomposing prime factors. Short division is often used to factor prime factors. First divide by prime numbers that can divide the composite number, keep dividing until the quotient is a prime number, and then write the divisor and quotient in the form of continuous multiplication.
⑶ Common factor (approximate) number
The common factors of several numbers are called the common factors of these numbers. The largest one is called the greatest common factor of these numbers.
Two numbers whose common factors are only 1 are called coprime numbers. Two numbers that are in a mutually prime relationship have the following situations: ① Relatively prime with any natural number;
② Two adjacent natural numbers are relatively prime;
③ When the composite number is not a multiple of a prime number, the composite number is relatively prime with the prime number ;
④ When the common divisor of two composite numbers is only 1, the two composite numbers are relatively prime. If any two of several numbers are relatively prime, it is said that these numbers are mutually prime.
If the smaller number is a divisor of the larger number, then the smaller number is the greatest common divisor of the two numbers.
If two numbers are relatively prime, their greatest common divisor is 1.
⑷ Common multiples
① The common multiples of several numbers are called the common multiples of these numbers. The largest one is called the greatest common multiple of these numbers.
The method to find the greatest common divisor of several numbers is: first use the common divisors of these numbers to divide continuously until the obtained quotient only has a common divisor of 1, and then multiply all the divisors together to find the product. This product is The greatest common divisor of these numbers.
② The common multiples of several numbers are called the common multiples of these numbers, and the smallest one among them is called the least common multiple of these numbers.
The method for finding the least common multiple of several numbers is: first divide by the common divisors of these numbers (or part of them) until they are relatively prime (or pairwise relatively prime), and then sum up all the divisors Multiply the quotients together to find the product. This product is the least common multiple of these numbers.
If the larger number is a multiple of the smaller number, then the larger number is the least common multiple of the two numbers.
If two numbers are relatively prime, then the product of the two numbers is their least common multiple.
The number of common divisors of several numbers is limited, but the number of common multiples of several numbers is infinite.
Properties and laws
(1) The law of constant quotient
The law of constant quotient: In division, the dividend and divisor are expanded or reduced by the same times at the same time, and the quotient remains unchanged.
(2) Properties of decimals
Properties of decimals: Add zero or remove zero at the end of the decimal, the size of the decimal remains unchanged.
(3) The movement of the decimal point position causes changes in the size of the decimal . Move the decimal point one place to the right, and the original number will be expanded by 10 times; move the decimal point two places to the right, and the original number will be expanded by 100 times; move the decimal point three places to the right. digit, the original number will be expanded 1000 times... . Move the decimal point one place to the left, and the original number will be reduced 10 times; move the decimal point two places to the left, and the original number will be reduced 100 times; move the decimal point three places to the left, The original number is reduced by 1000 times... . When the decimal point is moved to the left or to the right and there are not enough digits, "0" must be used to make up the digits.
(4) Basic properties of fractions
Basic properties of fractions: The numerator and denominator of a fraction are multiplied or divided by the same number (except zero), and the size of the fraction remains unchanged.
(5) The relationship between fractions and division . Dividend ÷ divisor = dividend/divisor . Because zero cannot be used as a divisor, the denominator of a fraction cannot be zero. . The dividend is equivalent to the numerator, and the divisor is equivalent to the denominator.
Operation rules
(1) The rules of the four arithmetic operations of integers . Integer addition:
The operation of combining two numbers into one number is called addition.
In addition, the number added is called the addend, and the number added is called the sum. The addend is the part number and the sum is the total number.
addend + addend = sum one addend = sum - another addend
. Integer subtraction:
It is known that the sum of two addends and one of the addends, the operation of finding the other addend is called subtraction.
In subtraction, the known sum is called the minuend, the known addend is called the subtrahend, and the unknown addend is called the difference. The minuend is the total number, and the subtrahend and difference are the partial numbers respectively.
addition and subtraction are inverse operations of each other.
3, integer multiplication :
The simple operation of finding the sum of several identical addends is called multiplication.
In multiplication, the same addends and the number of the same addends are called factors.The sum of the same addends is called the product.
In multiplication, 0 multiplied by any number gets 0. Multiplying 1 with any number gets any number.
One factor
In division, the known product is called the dividend, a known factor is called the divisor, and the required factor is called the quotient.
multiplication and division are the inverse operations of each other.
In division, 0 cannot be used as the divisor. Since 0 multiplied by any number is 0, any number divided by 0 cannot get a definite quotient.
dividend ÷ divisor = quotient divisor = dividend ÷ quotient dividend = quotient × divisor
. Power:
The operation of finding the product of several identical factors is called exponentiation. For example, 3 × 3 =32
(2) Four decimal arithmetic operations
1, decimal addition:
The meaning of decimal addition is the same as that of integer addition. It is an operation that combines two numbers into one number.
. Decimal subtraction:
The meaning of decimal subtraction is the same as that of integer subtraction. Given the sum of two addends and one of the addends, find the operation of the other addend.
. Multiplication of decimals:
The meaning of multiplying decimals by integers is the same as the meaning of multiplication of integers, which is a simple operation to find the sum of several identical addends; the meaning of multiplying a number by a pure decimal is to find the tenths and hundredths of the number. , a few thousandths...how much is it.
. Decimal division:
The meaning of decimal division is the same as that of integer division. It is the operation of finding the product of two factors and one of the factors to find the other factor.
(3) The four arithmetic operations of fractions
1. Fraction addition:
The meaning of fraction addition is the same as that of integer addition. It is an operation that combines two numbers into one number. . Subtraction of fractions: The meaning of
subtraction of fractions is the same as that of subtraction of integers. Given the sum of two addends and one of the addends, find the operation of the other addend. . Fractional multiplication:
The meaning of fractional multiplication is the same as that of integer multiplication, which is a simple operation to find the sum of several identical addends. . Fractional division:
The meaning of fractional division is the same as that of integer division. It is the operation of finding the product of two factors and one of the factors to find the other factor.
(4) Laws of operation
1, Laws of addition
⑴ Commutative law of addition:
When two numbers are added, the positions of the addends are exchanged, and their sum remains unchanged, that is, a+b=b+a.
⑵ Associative law of addition:
To add three numbers, add the first two numbers first, and then add the third number; or add the last two numbers first, and then add their sum to the first number. unchanged, that is (a+b)+c=a+(b+c).
. The law of multiplication
⑴ The commutative law of multiplication:
When two numbers are multiplied, their product remains unchanged at the position of the exchange factor, that is, a×b=b×a.
⑵ Associative law of multiplication:
To multiply three numbers, first multiply the first two numbers and then multiply by the third number; or first multiply the last two numbers and then multiply by the first number. Their The product remains unchanged, that is, (a×b)×c=a×(b×c).
⑶Distributive law of multiplication:
The sum of two numbers is multiplied by a number. You can multiply the two addends by this number respectively, and then add the two products, that is (a+b)×c=a×c +b×c.
⑷ Extension of the distributive law of multiplication:
The difference between two numbers is multiplied by one number. You can first multiply them by this number and then subtract them, that is (a-b) ×c=a×c-b×c
. The law of subtraction
⑴ To subtract several numbers from a number continuously, you can subtract the sum of all subtrahends from this number without changing the difference, that is, a-b-c=a-(b+c).
⑵ To subtract two numbers from one number continuously, you can subtract the second subtrahend first, and then subtract the first subtrahend, that is, a-b-c=a-c-b.
. The law of division operation
⑴ If a number is divided by two numbers continuously, it can be divided by the set of these two numbers, that is, a÷b÷c=a÷(b×c).
⑵ If a number is divided by two numbers continuously, it can be divided by the second divisor first, and then divided by the first divisor, that is, a÷b÷c=a÷c÷b.
, other
a-b+c=a+c-b
a-b+c=a+(b-c)
a÷b×c=a×c÷b
a÷b×c=a÷(b÷c)
. Change rule: In multiplication, one factor remains unchanged, the other factor expands (or shrinks) several times, and the product also expands (or shrinks) by the same multiple.
promotion: One factor is expanded by A times, the other factor is expanded by B times, and the product is expanded by AB times.
One factor reduces it by A times, the other factor reduces it by B times, and the product is reduced by AB times.
. Invariant quotient property: In division, the dividend and divisor are expanded (or reduced) by the same multiple at the same time, and the quotient remains unchanged. m≠0 a÷b=(a×m) ÷(b×m)=(a÷m) ÷(b÷m)
Generalization: The dividend is expanded (or reduced) A times, the divisor remains unchanged, and the quotient also expands ( or shrink) A times.
The dividend remains unchanged, the divisor expands (or shrinks) by A times, but the quotient shrinks (or expands) by A times.
can make some calculations simple by using the changing laws of products and the invariant properties of quotients. But be careful about the remainder in division with a remainder. For example: 8500÷200= You can divide by reducing the dividend and divisor by 100 times at the same time, that is, 85÷2=, the quotient remains unchanged, but the remainder 1 at this time is reduced by 100 times, so the original remainder should be 100.
(5) Calculation method
1, integer addition calculation rule:
The same digits are aligned, starting from the low digit, and whichever digit adds up to ten, advance one to the previous digit.
. Calculation rules for integer subtraction:
Align the same digits and add them up from the low digit. If the number in any digit is not enough to subtract, go back one digit from the previous digit and make ten, merge it with the number in the original digit, and then subtract.
. Calculation rules for integer multiplication:
First use the number in each digit of one factor to multiply the number in each digit of another factor. Use the number in each digit of the factor to multiply, and the ends of the multiplied numbers will be aligned. One digit, and then add up the multiplied numbers.
. Calculation rules for integer division:
Divide from the high digit of the dividend first. How many digits the divisor is depends on the first few digits of the dividend. If it is not enough to divide, look at one more digit. Which digit of the dividend is divided, the quotient is Which one is written above. If there is not enough quotient of 1 in any bit, "0" should be added to occupy the place. The remainder of each division must be less than the divisor.
. Decimal multiplication rules:
First calculate the product according to the calculation rules of integer multiplication, then look at how many decimals there are in the factors, count them from the right side of the product, and click the decimal point; if there are not enough digits, use "0" "Complement."
. Calculation rules for decimal division when the divisor is an integer:
First divide according to the rules of integer division. The decimal point of the quotient should be aligned with the decimal point of the dividend; if there is still a remainder at the end of the dividend, add "0" after the remainder, and then Continue to remove.
. Calculation rules for division when the divisor is a decimal:
first moves the decimal point of the divisor to make it an integer, and also moves the decimal point of the divisor a few places to the right (if there are not enough digits, add "0"), and then follow the division method when the divisor is an integer. Calculate according to the rules.
. Calculation method of adding and subtracting fractions with the same denominator:
To add and subtract fractions with the same denominator, only add and subtract the numerators, leaving the denominator unchanged.
. Calculation method of adding and subtracting fractions with different denominators:
first use common denominators, and then calculate according to the rules of adding and subtracting fractions with the same denominator.
0. Calculation method of adding and subtracting mixed fractions:
Add and subtract the integer parts and fractional parts respectively, and then combine the resulting numbers.
1. Calculation rules for fraction multiplication:
When multiplying a fraction by an integer, use the product of the numerator of the fraction and the integer as the numerator, and the denominator remains unchanged; when multiplying a fraction by a fraction, use the product of the numerators as the numerator, and the product of the denominators as the denominator. .
2. Calculation rules for fraction division:
Number A divided by number B (except 0) is equal to the reciprocal of number A times number B.
(6) The order of operations
1, the order of operations of the four arithmetic operations on decimals is the same as the order of the four arithmetic operations on integers.
2. The order of operations of the four arithmetic operations on fractions is the same as the order of the four arithmetic operations on integers. . Mixed operations without parentheses: same-level operations are performed from left to right; two-level operations are performed first, multiplication and division, and then addition and subtraction.. Mixed operations with parentheses: first calculate the items inside the parentheses, then calculate the items inside the square brackets, and finally calculate the items outside the brackets. , first-level operations: addition and subtraction are called first-level operations. . Second-level operations: Multiplication and division are called second-level operations.
