﻿ Division — Math18

Math18

# Division

Content

## What is division?

Division is one of the four basic operations of arithmetic that consists in finding out how many times a number (divisor) is contained in another number (dividend). The division can be considered an operation equivalent to the subtraction since the divided number can be set as equivalent in a subtraction, for example: 6/2 = 3 corresponding to 6 - 2 = 4 (1st subtraction), 4 - 2 = 2 (2nd subtraction) and 2 - 2 = 0 (3rd subtraction), therefore, it is concluded that we have 3 subtractions and it is the equivalent of dividing 6/2 = 3. Division, of the Latin division, is the action and the result to divide

### Symbol or sign of division

The representation or sign of the division that is known as “between”, is by means of a diagonal (/) or an obelus (÷), in some cases it is represented by a colon (:).

## Parts of the division

When performing a division operation, 4 important elements are considered:

• Divisor: It is the number or amount by which we will divide, according to the amount indicated by the dividend.
• Dividend: It is the amount that we want to distribute and for which we will make the division.
• Quotient: It is the result of the division
• remainder: The residue or also known as the remainder, is the number or surplus number of the division.
3 ← Quotient Divisor → 4 12 ← Dividend                    -12                       0 ← Remainder

Other ways of representing the division considering the dividend (D) and the divisor (d):

D / d          D ÷ d          D : d

## Division Properties

There are different basic properties that are met in a division:

• Division between 1: Any number divided by 1 will be the same number, example: 4/1 = 4, 12/1 = 12.
• Divide the 0: Any number that divides 0 will be zero, example: 0/5 = 0, 0/12 = 0.
• Division between 0: Any number divided by zero is considered an infinite number (inf).

To verify that the division between 0 is infinite we can use the subtraction as equivalent, where 6/0 = inf, therefore 6 - 0 = 6 (1st subtraction), 6 - 0 = 6 (2nd subtraction), 6 - 0 = 6 (3rd subtraction),…, 6 - 0 = 6 (infinite subtraction), because when subtracting zero the value of zero is never obtained in the result, it is considered that any number divided by zero gives an infinite value result.

## How can we divide?

The best way to learn to divide is to use objects, in the following example you have 4 balls and 2 boats or containers in which you want to accommodate the same amount of balls in each container, therefore, you must divide 4 balls by 2 boats equivalent to 4/2 = 2, that means that each boat or container must have 2 balls.

There are different learning methods for the realization of divisions, among these methods we can find especially two, which are used for numbers of small quantities and the other method for numbers of large quantities.

• Direct division: It is used in divisions of a small amount, as experience is gained, the ease of this method for larger numbers increases. It should be considered that for a person who is learning 8/4 mathematics it can be a bit confusing, but the purpose is to gradually raise the difficulty. This method helps the learning of mental calculation.

Examples:

4 / 4 = 1
8 / 2 = 4
6 / 3 = 2
9 / 3 = 3

Exercise:

A) 4 / 4 = ?
B) 4 / 2 = ?
C) 8 / 4 = ?
D) 2 / 1 = ?
• Division by parts: It is a method for the realization of large divisions, it is important to have an order in the arrangement of numbers since placing them in an inappropriate position can generate an error in the division. Considering exercise 52/2 of the examples below, we first see if the first unit that is 5 is divisible by 2, when doing the operation we have 5/2 = 2 and we have a residue of 1, then the residue "1" we will add the number "2", now we must divide 12/2 = 6 and we have a residue of "0" therefore it is considered as exact division.
26 2 52   -4    12   -12      0

Examples:

4 3 12   -12      0
6 3 18   -18      0
12 3 36   -3    06     -6      0

Exercises:

A)
2 18 = ?
B)
8 32 = ?
C)
2 36 = ?
D)
5 65 = ?

In some divisions we will have the case of the residue. What is the remainder? It corresponds to the number that is left over from the division, in some cases we can extend the division by adding a decimal point and in this way we can obtain a residual of 0, but in other cases it can be a constant residue, in the division section with decimal point The procedure is explained.

The exact divisions are those in which the residual is equal to 0 and on the other hand the inaccurate divisions are those in which there is a non-zero residual.

Examples:

4 3 14   -12      2
6 4 25   -24      1
3 9 33   -27      6
8 7 62   -56      6

The red number represents the remainder of the division.

Exercise:

How much is the remainder of the following operations? Ramainder = ?
A)
2 17 = ?
B)
4 9 = ?
C)
5 34 = ?
D)
9 80 = ?

## Decimal Division

The division with decimals can be generated because the divisor or dividend has decimal numbers or because the residual is different from zero.

The example of division 23/8 will be followed:

We perform the division of 23/8 as the steps previously seen:
2 8 2 3   -1 6       7
Since divisor 8 cannot divide 7, a decimal point and a zero are added to the right of the dividend, zero is placed in the same way to one side of the residual.
2 . 8 2 3 . 0   -1 6       7   0
Now we look for a number that multiplied by 8 results in 70. The number 8 is the factor that multiplied by 8 is equal to 64, this factor is considered because 64 is less than 70.
2 . 8 8 2 3 . 0   -1 6       7   0      -6   4       0   6
Factor 8 passes to the right of the quotient and the decimal point is placed in the same position as in the dividend.
The steps are repeated again, in the following box you will find the complete example.

Examples:

2 . 8 7 5 8 2 3 . 0 0 0   -1 6       7   0      -6   4       0   6 0           -5 6            0 4 0              -4 0            0 0 0

The red number represents the zeroes added and the quotient to the right of the point. The division concludes when the residue is zero.

Exercise:

A)
2 17 = ?
B)
4 9 = ?
C)
5 34 = ?
D)
9 80 = ?

Note: As you can see in exercise D, the division cannot be exact because the waste is never going to be 0.

### Division with decimals in the divisor

When you have decimals in the divisor it is advisable to multiply by a multiple of 10 until you get the divisor as an integer, for example:

1.86 x 100 = 186
1.5 x 10 = 15
1.964 x 1000 = 1964
It is important to consider that the value multiplied in the divisor must also be multiplied in the dividend, not performing the operation would affect the result. A better way to visualize is to accommodate as a fraction.
As an example you have 30 / 1.5:
1.5 3 0
Multiply by 10 to convert the divisor to an integer.
30 x 10 = 300
1.5 x 10 = 15
15 3 0 0
It is now possible to perform the division as the steps previously seen.
2 0 15 3 0 0     -3 0         0 0

### Division with decimal places in the dividend

Having a decimal point in the dividend should only be considered to raise the point to the quotient, for example 4.5/15

0 . 3 15 4 . 5     -4   5           0

### Division with decimals in the divisor and in the dividend

For this case the important thing is to have the divisor as integer, that facilitates the resolution of the problem. For example: 8.61/2.1

Accommodated in the form of division:

2.1 8 . 6 1

Multiplying by 10 you have 8.61 x 10 = 86.1 y 2.1 x 10 = 21

Now it is possible to solve the division in a simple way:

4 . 1 21 8 6 . 1     -8 4         2   1        -2   1              0

## Check a division

The division can be checked with its inverse mathematical operation, this is the multiplication but considering the residual as sum. To verify a division, the quotient is multiplied by the divisor and then the residual is added, resulting in the dividend, indicating that the division was correct.

Dividend = (Quotient x Divisor) + Remainder

Example:

23 /
8
= 2.875
(2.875 x 8) + 0
= 23

This operation checks the result of the division is correct.

## Division of fractions

When in a division the residue is not zero, the division can be expressed as a fraction. To solve a division of fractions there is a slightly more complex procedure to which corresponds Sum, Subtraction or Multiplication of fractions, but understanding how the procedure is, you can see how simple it can be.

Example:

a)
1 / 2
= 0.5
b)
2 / 3 / 4 / 6
=
12 / 12
= 1