# Translations of Encyclopedia about Mathematics

## Division

As you can see, division is the opposite of multiplication, for which reason multiplication may also serve as a controlling mechanism to make sure we performed a division correctly.

Multiplication: 7 . 4 = 28
Division: 28: 4 = 7

The number shown before the division symbol is called the dividend and the number after the division symbol the factor or divisor. The result is called the quotient. In the above example, the dividend would be the number 28, the factor the number 4 and the quotient the number 7.

If we want to divide large numbers without the use of a calculator, the following approach can be used:

702 :13 = 54

- 65 13 fits into the number 70 5 times, although 5 is left over

52

- 52 13 fits into 52 4 times but nothing is left over

0

The first row shows the function at hand, 702/13, where the quotient is written after the equal sign.

The first step is to determine how many times we can multiply the number 13 to give us a value smaller than 70. In our case it is the number 5.

The second row contains the result of multiplying 13 and 5, which is 65, which we subtract from 70 (make sure that the remainder is smaller than the factor, which in this case is 13).

The third row shows the result of the subtraction (70-65) followed by the next number (following the 70) in the dividend (in our case 2). We then continue in the same manner, but this time find that 13 fits into 52 exactly 4 times, in which case there is no more remainder (the value 0 written at the bottom). Once the value at the end is 0, it means we have completed our calculation.

If the result of division is not 0, it means that some remainder is left over, which we can include in the final result in the form of a fraction:

518 : 35 = 14 + 28/35

-35

168

-140

28

or in the form of a decimal number:

518 : 35 = 14, 8

-35

168

-140

280 (the added 0 is marked in the resulting value after the decimal point)

-280

0

If the calculation does not end in the first position after the decimal point, another 0 is added to the next remainder to give the next, second number after the decimal point. This process continues until we get our final result of 0. However, it is most often suitable to stop at some point and round up or down some number after the decimal point.

The following rules apply for multiplication, provided a and b are real numbers:

(+a) . (+b) = + (a . b)

(+a) . (-b) = - (a.b)

(-a) . (+b) = - (a.b)

(-a) . (-b) = + (a.b)

Simply said: "+" times "+" equals"+"

"+" tims "-" equals "-"

"-" times "-" equals "+"

The same applies for divisions.

Taking a=4 and b=3 as an example, when applying the rules of multiplication, we get:

(+4) . (+3) = + (4.3) =+12
(+4) . (- 3) = - (4.3) = - 12
(-4) . (+3) = - (4.3) = - 12
(-4) . (- 3) = + (4.3) = +12

If more arithmetical functions appear in a particular equation, first multiplication and division must be performed and only then addition and subtraction. If, however, there are brackets in an equation, those functions within the brackets must be performed first.

It therefore applies that:

Brackets have precedence before multiplication and division, and multiplication and division have precedence before addition and subtraction.

Examples:

4 + 6 . 3 = 4 + 18 = 22
12 : 2 + 7 . 3 = 6 + 21 = 27
(5 + 3) : 2 = 8 : 2 = 4

The so-called rules of division are important for every type of division and for disollution to primary numbers.

Natural numbers are those which are:

divisible by 2 if even, such as 12, 26 or 76

divisible by 3 if the sum of its digits are divisible by 3, such as 162 (1 + 6 + 2 = 9, 9 is divisible by 3)
divisible by 4 if its last two digits form a number divisible by 4, such as 65424
divisible by 5 if its last number is 0 or 5, such as 98755
divisible by 8 if its last three digits form a number which is divisible by 8, such as 8745160
divisible by 9 if the sum of its digits are divisible by 9, such as 59886: 5+9+8+8+6 = 36
divisible by 10 if its last digit is a 0
divisible by 25 if its last two digits are divisible by 25, such as 6575

divisible by 100 if its last two digits are 0, such as 6231400.

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