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Arithmetic operation

Mathematical terminology

This entry is made by Compilation and application of scientific encyclopedia entries of "Science Popularization China" to examine.

Arithmetic operation, namely "four operations", is addition 、 subtraction 、 multiplication and division A general term for the four operations. It is usually real number or complex Conducted. Two numbers belonging to a number set can determine the third number of the number set through arithmetic operation. In computers, arithmetic operations also include computing absolute value , "negation" and logical operation "comparison".^[1] (A few books are called Power 、 Prescription It also belongs to arithmetic operation).

Chinese name: Arithmetic operation
Foreign name: arithmeticaloperation

Discipline: mathematics
Alias: Four arithmetic operations

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definition

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Arithmetic operation It is called operation for short. It refers to the process of calculating formula problems or formulas according to the specified rules and order and finding the results. Including: addition, subtraction, multiplication, division, power, and square root. Among them, addition and subtraction are primary operations, multiplication and division are secondary operations, and power and root are tertiary operations. In a formula, if there are several levels of operation, the high-level operation should be performed first, and then the lower level operation. For example: 3+22 × 4=3+4 × 4=3+16=19; If only sibling operation exists; Then proceed from left to right; If there are brackets in the formula, the calculation shall be carried out according to the above rules. For example: (3+2) 2 × 4=52 × 4=100. There is a slight difference between operation and calculation. Calculation refers to the calculation of the number in the horizontal formula according to the operation symbol and the specified order. The result can be obtained directly according to the operation rule, or by oral calculation or other simple methods. Operation refers to the process of obtaining results.^[2]

Definition of each operation

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addition The operation of combining two numbers into one number.

subtraction : When the sum of two addends and one of them is known, the operation of finding another addend is performed.

multiplication : The operation of multiplying two numbers. Including:

(1) A number multiplied by an integer is a simple operation for finding the sum of several identical addends;

(2) Multiplying a number by a decimal is to find the number of tenths, percentages, thousandths;

(3) Multiplying a number by a fraction is to find the fraction of the number.

division : Given the product of two factors and one of them, calculate the operation of the other factor.^[3]

Power (Powers of Numbers): Find n numbers of the same factor product The operation of is called the power, and the result of the power is called power 。 stay

Medium,

It's called the base number, and n is called index ，

pronounce as

To the nth power of.

regard as

When the result is to the nth power of, it can also be read as

To the nth power of. The second power is also called the square, and the third power is also called the cube. Any power of a positive number is a positive number; The odd power of a negative number is negative, and the even power of a negative number is positive.

Prescription (Roots of Numbers): Generally, if the square of a number is equal to

This number is called

The square root of (also called quadratic root), in other words, if

, then x is called

The square root of.

Generally speaking, a positive number has two square root , these two square roots are mutually Inverse number , the square root of zero is zero. In formula

Medium,

Is called the squared number, and 2 is called Root index 。

Positive number

The positive square root of

Of arithmetic square root ； The square root of zero is also called the arithmetic square root of zero, so the arithmetic square root of zero is still zero.^[4]

Relationship of various parts

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Multiplication is a simple operation of addition, and division is a simple operation of subtraction.

Subtraction and addition are mutually inverse operations, and division and multiplication are mutually inverse operations.

Addend+addend=sum;

Subtracted number minus=difference;

One addend=and - another addend;

Subtract=minuend - difference;

Subtracted number=difference+subtracted number;

Factor × factor=product;

One factor=product ÷ another factor;

Dividend ÷ divisor=quotient;

Divider=dividend ÷ quotient;

Dividend=quotient × divisor^[3]

Arithmetic algorithm

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The rule of addition and subtraction

integer

(1) Same digit alignment;

(2) Counting from one digit;

(3) In addition, if the number is more than a few tens, move to the higher one; When the subtraction is not enough, it will be subtracted from the higher digit by 1 when 10 is added to the current digit.

decimal

(1) Decimal point alignment (that is, the same digit alignment);

(2) Calculate according to the rule of integer addition and subtraction;

(3) Align the decimal point on the horizontal line in the result, and point to the decimal point.

fraction

(1) Add and subtract fractions with the same denominator, the denominator remains unchanged, and only the numerators are added and subtracted;

(2) Add and subtract fractions with different denominators, first General distribution , and then calculate according to the rule of addition and subtraction of fractions with the same denominator;

(3) The result is not the simplest fraction Approximation become Minimalist fraction 。

The rule of multiplication

integer

(1) Starting from the single digit, multiply the first factor by the number on each of the second factor in turn;

(2) Multiply the number in the bit of the second factor, and the last bit of the number will be aligned with the bit of the second factor;

(3) Add up the numbers multiplied several times.

decimal

(1) Calculate the product according to the rule of integer multiplication;

(2) Look at the number of decimal places in the factor, and count the decimal points from the right side of the product.

fraction

(1) Fraction multiplies fraction, using the product of numerator multiplication as numerator, and the product of denominator multiplication as denominator;

(2) The integer is regarded as a false fraction whose denominator is 1;

(3) If you can reduce the score, you should first determine the score.

The law of division

integer

(1) Divide from the high order of the dividend;

(2) If the divisor is a few digits, first look at the first few digits of the divisor. If it is not enough to divide, then look at one more digit;

(3) The business shall be written on the top of each other;

(4) The remainder of each division must be smaller than the divisor;

(5) After finding the highest quotient, if the digit of the dividend is less than the quotient 1, write 0 in the digit.

decimal

(1) When the divisor is an integer, it shall be calculated by integer division, and the decimal point of quotient shall be aligned with the decimal point of the divisor;

(2) When the divisor is a decimal, it is first converted to the decimal division with integer divisor, and then calculated according to the decimal division with integer divisor.

fraction

A number divided by B number (except 0) is equal to the reciprocal of A number multiplied by B.

Operational property

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Nature of addition

From the commutative law and associative law of addition, we can get that: when several addends are added, the position of addends can be exchanged at will; Or you can add several addends first and then add them with other addends, and their sum remains the same. For example: 34+72+66+28=(34+66)+(72+28)=200.

Subtraction Properties

① Subtracting the sum of two numbers from one number is equal to subtracting each addend from the sum in turn. For example, 134 - (34+63)=134 - 34 - 63=37;

② Subtracting the difference between two numbers from a number is equal to subtracting the minuend of the difference from the number and adding the subtraction. For example, 100 - (32 - 15)=100 - 32+15=68+15=83;

③ Subtract a number from the sum of several numbers. You can select one of the addends to subtract this number, and then add it with the rest of the addends. For example: (35+17+29) - 25=35 - 25+17+29=56;

④ To subtract several numbers from a number consecutively, you can add all the subtractors first, and then subtract the sum of the subtractions from the minuend. For example, 276 － 115 － 85=276 － (115+85)=76.

Properties of multiplication

① Multiply the product of several numbers by a number, and let any factor in the product multiply this number, and then multiply with other numbers. For example: (25 × 3 × 9) × 4=25 × 4 × 3 × 9=2700;

② When the difference between two numbers is multiplied by a number, the minuend and the subtractor can be multiplied by this number respectively, and then the resulting product can be subtracted. For example: (137 - 125) × 8=137 × 8-125 × 8=96.

Nature of division operation

① If a number is divided (or multiplied) by a number, and then multiplied (or divided) by the same number, the number remains unchanged. For example: 68 ÷ 17 × 17=68 (or 68 × 17 ÷ 17=68);

② When a number is divided by the product of several numbers, you can use this number to divide each factor in the product in turn. For example: 320 ÷ (2 × 5 × 8)=320 ÷ 2 ÷ 5 ÷ 8=4;

③ The quotient of a number divided by two numbers is equal to the number divided by the dividend in the quotient and multiplied by the divisor in the quotient. For example: 56 ÷ (8 ÷ 4)=56 ÷ 8 × 4=28;

④ Divide the product of several numbers by a number. You can divide any factor in the product by this number and multiply it by other factors. For example: 8 × 72 × 4 ÷ 9=72 ÷ 9 × 8 × 4=256;

⑤ Divide the sum of several numbers by one number. You can first divide each addend by this number, and then add each quotient. For example: (24+32+16) ÷ 4=24 ÷ 4+32 ÷ 4+16 ÷ 4=18;

⑥ The difference between two numbers divided by a number can be subtracted from the quotient of the minuend divided by the number. For example: (65-39) ÷ 13=65 ÷ 13-39 ÷ 13=2.