Digital algorithm
(A) the rules of arithmetic operation of four integers
1, integer addition:
The operation of combining two numbers into one number is called addition.
In Djaafari, the added number is called addend, and the added number is called sum. The appendix is a partial figure, and the sum is the total.
Appendix+Appendix = and one addend = and-another addend.
2, integer subtraction:
Given the sum of two addends and one of them, the operation of finding the other addend is called subtraction.
In subtraction, the known sum is called the minuend, the known addend is called subtraction, and the unknown addend is called difference. The minuend is the total number, and the subtraction and difference are the partial numbers respectively.
Addition and subtraction are reciprocal operations.
3, integer multiplication:
The simple operation of finding the sum of several identical addends is called multiplication.
In multiplication, the same addend and the same number of addends are called factors. The sum of the same addend is called product.
In multiplication, multiplying 0 by any number will get 0. 1, then multiply any number by any number.
A factor? A factor = product. A factor = product? Another factor
4, integer division:
Given the product of two factors and one of them, the operation of finding the other factor is called division.
In division, the known product is called dividend, the known factor is called divisor, and the calculated factor is called quotient.
Multiplication and division are reciprocal operations.
In division, 0 cannot be divided. Because 0 is multiplied by any number to get 0, any number divided by 0 can't get a definite quotient.
Dividend? Divider = quotient divisor = dividend? Business dividend = business? divisor
5. Multiplier:
The operation of finding the product of several identical factors is called power. Like 3? 3 =32
(2) Four decimal places operation
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.
2. Decimal subtraction:
Decimal subtraction and integer subtraction have the same meaning. Know the sum of two addends and one of them, and find the other addend.
3. Decimal multiplication:
The meaning of decimal multiplication by integer is the same as that of integer multiplication, 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 out a few tenths, a few percent and a few thousandths of this number.
4, fractional division:
The significance of fractional division is the same as integer division, that is, knowing the product of two factors and one of them, and finding the other factor.
(3) Four Fractions Operation
1, fractional addition:
Fractional addition and integer addition have the same meaning. It is an operation that combines two numbers into one number.
2. Fractional subtraction:
The significance of fractional subtraction is the same as that of integer subtraction. The operation of finding the other addend by knowing two addends and one of them.
3. Fractional multiplication:
The significance of fractional multiplication is the same as that of integer multiplication, which is a simple operation to find the sum of several identical addends.
4, fractional division:
Fractional division has the same meaning as integer division. It is an operation to find the other factor by knowing the product of two factors and one of them.
(4) Operation law
1, addition algorithm
Additive commutative law;
When two numbers are added, the positions of addends are exchanged, and their sum is unchanged, that is, A+B = B+A.
(2) Additive associative law:
Add three numbers, first add the first two numbers, then add the third number; Or add the last two numbers first, and then add the first number, and their sum remains the same, that is, (a+b)+c=a+(b+c).
2. Law of multiplication operation
(1) multiplicative commutative law:
When two numbers are multiplied, the position of the exchange factor remains the same, that is, a? b=b？ Answer.
(2) Multiplicative associative law:
Multiply three numbers, first multiply the first two numbers and then multiply the third number; Or multiply the last two numbers first, and then multiply them with the first number, and their products remain the same, that is, (a? b)？ c=a？ (b？ C) carry out the test.
(3) Law of multiplication and distribution:
When 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.
(4) Extension of the law of multiplication and distribution:
When the difference between two numbers is multiplied by a number, you can multiply this number separately and then subtract it, that is, (a-b)? c=a？ c-b？ c
3. Law of subtraction operation
(1) If you subtract several numbers from a number continuously, you can subtract the sum of all the subtractions of this number, and the difference remains unchanged, that is, a-b-c=a-(b+c).
(2) If a number subtracts two numbers continuously, you can subtract the second subtraction first and then the first subtraction, that is, A-B A-B-C = A-C-B C-B.
4. Division algorithm
(1) A number is divisible by two numbers in a row and can be divisible by the set of these two numbers, that is, a? b？ c=a？ (b？ C) carry out the test.
(2) If a number is divisible by two numbers in succession, it can be divisible by the second divisor first and then by the first divisor, that is, a? b？ c=a？ c？ B.
5. Others
a-b+c=a+c-b
a-b+c=a+(b-c)
Answer? b？ c=a？ c？ b
Answer? b？ c=a？ (b？ c)
6. Variation law of product: In multiplication, if one factor remains unchanged and another factor is expanded (or reduced) several times, the product is also expanded (or reduced) by the same multiple.
Summary: One factor magnifies factor A, another factor magnifies factor B, and the product magnifies factor AB.
One factor subtracts one factor, another factor subtracts B, and the product subtracts AB.
7. Quotient invariance: 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 enlarges (or reduces) a factor, and the quotient enlarges (or reduces) a factor, while the divisor remains the same.
The dividend is constant, the divisor is enlarged (or reduced) by a factor, but the quotient is reduced (or expanded) by a factor.
By using the changing law of product and the invariant law of quotient, some calculations can be simplified. But pay attention to the remainder in the division with remainder. Such as: 8500? 200= Divider and divisor can be reduced by 100 times at the same time, that is, 85? 2=, the quotient is unchanged, but the remainder 1 minus 100, so the original remainder should be 100.
(5) Calculation method
1, integer addition calculation rules:
The same numbers are aligned from the low order. When the numbers add up to ten, they will advance to the previous number.
2, integer subtraction calculation rules:
The same numbers are aligned from the lower number. If the number of digits is not reduced enough, the number of digits from the last digit is reduced by ten, and then it is combined with the number of digits in the radix and then reduced.
3, integer multiplication calculation rules:
First, multiply the number on each bit of one factor by the number on each bit of another factor, then multiply the number on which bit of the factor, align the end of the multiplied number with which bit, and then add the multiplied numbers.
4, integer division calculation rules:
Divide from the high order of the dividend, which is a few digits, depending on the first few digits of the dividend; If the division is not enough, look at another place and the quotient is written on the dividend. If any number is not quotient 1, a "0" placeholder should be added. The remainder of each division should be less than the divisor.
5, decimal multiplication rule:
First, calculate the product according to the calculation rules of integer multiplication, and then look at the factor * * *, how many decimal places there are, just count a few from the right side of the product and point to the decimal point; If the number of digits is not enough, make up with "0".
6, divisor is integer decimal division calculation rules:
First of all, according to the law of integer division, the decimal point of quotient should be aligned with the decimal point of dividend; If there is a remainder at the end of the dividend, add "0" after the remainder to continue the division.
7, divisor is the calculation rule of fractional division:
First, move the decimal point of the divisor to make it an integer, then move the decimal point of the divisor to the right by several digits (the number of digits is not enough, make up "0"), and then calculate according to the division rule that the divisor is an integer.
8, with denominator fraction addition and subtraction calculation method:
Add and subtract fractions with denominator, only add and subtract numerators, and the denominator remains the same.
9, different denominator fraction addition and subtraction calculation method:
Divide first, and then calculate according to the addition and subtraction law of fractions with the same denominator.
10, with the calculation method of fractional addition and subtraction:
Add and subtract the integer part and the decimal part respectively, and then combine the obtained numbers.
1 1, calculation rules of fractional multiplication:
Fractions are multiplied by integers, and the product of the multiplication of fractions and integers is taken as the numerator, and the denominator remains unchanged; Fractions are multiplied by fractions, the product of numerator multiplication is numerator, and the product of denominator multiplication is denominator.
12, calculation rules of fractional division:
The number A divided by the number B (except 0) is equal to the reciprocal of the number A multiplied by the number B.
(6) Operation sequence
1 and decimal arithmetic operate in the same order as integer arithmetic.
2. The operation sequence of fractional four operations is the same as that of integer four operations.
3. Mixed operation without brackets: the operations at the same level are operated from left to right in turn; Two-stage operation calculates multiplication and division first, and then addition and subtraction.
4. Mixed operation with brackets: first calculate what is in brackets, then calculate what is in brackets, and finally calculate what is outside brackets.
5, the first level of operation: addition and subtraction is called the first level of operation.
6. Secondary operation: Multiplication and division are called secondary operations.
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