maths > integers

Integer Division : First Principles

what you'll learn...

overview

This page introduces division of integers as -- one number, dividend, is split into the number of parts given by the second number, divisor. The count or measure of one part is the result, quotient. And the remaining count of dividend, that could not be split, is the remainder of the division.

The definition of division in first principles form the basis to understanding simplified procedure for division of large numbers.

split

In whole numbers, 6÷2$6 \div 2$ means: dividend 6$6$ is split into 2$2$ equal parts and one part is put in.

In integers, 2$2$ and 2$- 2$ are understood as
received:2=2$\textrm{\left(r e c e i v e d\right\rangle} 2 = 2$ and $\textrm{g i v e n : 2} = - 2$.

It is also called $\textrm{\left(a l i g \ne d\right\rangle} 2 = 2$ and $\textrm{o p p o s e d : 2} = - 2$

Integers are "directed" whole numbers.

A whole number division represents splitting the dividend into divisor number of parts and one part is put-in.

In integers,

•  positive divisor represents: one part is put-in

•  negative divisor represents: one part is taken-away

This is explained with an example in the coming pages.

put-in a received part

A girl has a box of candies. The number of candies in the box is not counted. But, she maintains a daily account of how many are received or given.

$6$ received is split into $2$ equal parts. In the box, one part of that is put-in. (To understand this : $6$ candies received is shared with her brother and only her part is put in the candy-box.)

The numbers in the integer forms are $\textrm{\left(r e c e i v e d\right\rangle} 6 = 6$ and $\textrm{\left(r e c e i v e d\right\rangle} 2 = 2$.

The number of candies received is
$\textrm{\left(r e c e i v e d\right\rangle} 6 = 6$ is split into $2$ parts and one part $\textrm{\left(r e c e i v e d\right\rangle} 3 = 3$ is put-in
$6 \div 2 = 3$

put-in a given part

Considering the box of candies and the daily account of number of candies received or given.

$6$ given is split into $2$ equal parts. In the box, one part of {$2$ equal parts of $6$ given} is put-in. (Her brother and she gave $6$ candies and only her part is reflected for her number.)

The numbers in the integer forms are $\textrm{\left(g i v e n\right\rangle} 6 = - 6$ and $\textrm{\left(r e c e i v e d\right\rangle} 2 = 2$. The number of candies received is
$\textrm{\left(g i v e n\right\rangle} 6 = - 6$ is split into $2$ parts and one part $\textrm{\left(g i v e n\right\rangle} 3 = - 3$ is put-in
$\left(- 6\right) \div 2 = - 3$

Considering division of $\left(- 6\right) \div 2$. The numbers are given in integer form. The numbers in directed whole numbers form are $\textrm{\left(g i v e n\right\rangle} 6$ and $\textrm{\left(r e c e i v e d\right\rangle} 2$.

The Division is explained as
$\textrm{\left(g i v e n\right\rangle} 6 = - 6$ is the dividend
$\textrm{\left(r e c e i v e d\right\rangle} 2$ is divisor

Division is dividend split into divisor number of parts and one part is put-in.
$- 6$ split into $2$ parts is $- 3$ and $- 3$. One part of that is $- 3$.
Thus the quotient of the division is $= \textrm{\left(g i v e n\right\rangle} 3$.

The same in integer form
$= \left(- 6\right) \div 2$
$= - 3$

take-away a received part

Considering the box of candies and the daily account of number of candies received or given.

$6$ received is split in $2$ equals part of which one part is to be taken-away. From the box, one part of {$2$ equal part of $6$ received} is taken-away. (Her brother and she returned $6$ candies that was received earlier and only her part is reflected for her number.)

The numbers in the integer forms are $\textrm{\left(r e c e i v e d\right\rangle} 6 = 6$ and $\textrm{\left(g i v e n\right\rangle} 2 = - 2$.

The number of candies received is
$\textrm{\left(r e c e i v e d\right\rangle} 6 = 6$ is split into $2$ parts and one part $\textrm{\left(r e c e i v e d\right\rangle} 3 = 3$ is taken away, which is $\textrm{\left(g i v e n\right\rangle} 3 = - 3$
$6 \div \left(- 2\right) = - 3$

Considering division of $6 \div \left(- 2\right)$. The numbers are given in integer form. To understand first principles of division, let us convert that to directed whole numbers form $\textrm{\left(r e c e i v e d\right\rangle} 6$ and $\textrm{\left(g i v e n\right\rangle} 2$.

The Division is explained as
$\textrm{\left(r e c e i v e d\right\rangle} 6 = 6$ is the dividend
$\textrm{\left(g i v e n\right\rangle} 2 = - 2$ is divisor

Division is dividend split into divisor number of parts and one part is taken away since divisor is negative.
$6$ split into $2$ parts is $3$ and $3$. One part of that is $3$. Since divisor is negative, one part $3$ is taken-away. $\textrm{\left(r e c e i v e d\right\rangle} 3$ taken away is $\textrm{\left(g i v e n\right\rangle} 3$.
Thus the quotient of the division is $= \textrm{\left(g i v e n\right\rangle} 3$.

The same in integer form
$= 6 \div \left(- 2\right)$
$= - 3$

take away a given part

Considering the box of candies and the daily account of number of candies received or given.

$6$ given is split into $2$ equal parts of which one part is to be taken-away. In the box, a part of {$2$ equal part of $6$ given} is taken-away. (Her brother and she got back $6$ candies which were given earlier and only her part is reflected for her number.)

The numbers in the integer forms are $\textrm{\left(g i v e n\right\rangle} 6 = - 6$ and $\textrm{\left(g i v e n\right\rangle} 2 = - 2$.

The number of candies received is
$\textrm{\left(g i v e n\right\rangle} 6 = - 6$ is split into $2$ parts and one part $\textrm{\left(g i v e n\right\rangle} 3 = - 3$ is taken-away, which is $\textrm{\left(r e c e i v e d\right\rangle} 3 = + 3$
$\left(- 6\right) \div \left(- 2\right) = + 3$

Considering division of $\left(- 6\right) \div \left(- 2\right)$. The numbers are given in integer form. To understand first principles of division, let us convert that to directed whole numbers form $\textrm{\left(g i v e n\right\rangle} 6$ and $\textrm{\left(g i v e n\right\rangle} 2$.

The Division is explained as
$\textrm{\left(g i v e n\right\rangle} 6 = - 6$ is the dividend
$\textrm{\left(g i v e n\right\rangle} 2 = - 2$ is divisor

Division is dividend split into divisor number of parts and one part is is taken away since the divisor is negative.
$- 6$ split into $2$ parts is $- 3$ and $- 3$. One part of that is $- 3$. Since divisor is negative, one part $- 3$ is taken-away. $\textrm{\left(g i v e n\right\rangle} 3$ taken away is $\textrm{\left(r e c e i v e d\right\rangle} 3$.
Thus the quotient of the division is $= \textrm{\left(r e c e i v e d\right\rangle} 3$.

The same in integer form
$= \left(- 6\right) \div \left(- 2\right)$
$= 3$

The summary of integer division illustrative examples:

•  $6 \div 2 = 3$
$6$ received split into $2$ parts and one part is put-in = $3$ received

•  $\left(- 6\right) \div 2 = - 3$
$6$ given split into $2$ parts and one part is put-in = $3$ given

•  $6 \div \left(- 2\right) = - 3$
$6$ received split into $2$ parts and one part is taken-away = $3$ given

•  $\left(- 6\right) \div \left(- 2\right) = 3$
$6$ given split into $2$ parts and one part is taken-away = $3$ received

The above is concise form to capture the integer division in first principles.

revising

The division $7 \div 3$ is understood as $\textrm{\left(r e c e i v e d\right\rangle} 7$ is split into $3$ equal parts and one part is put-in (positive divisor). The remainder is what is remaining in $\textrm{\left(r e c e i v e d\right\rangle} 7$.

The result is quotient $2$ and remainder $1$

This is verified with $2 \times 3 + 1 = 7$ (quotient multiplied divisor + remainder = dividend )

The division $\left(- 7\right) \div 3$ is understood as $\textrm{\left(g i v e n\right\rangle} 7$ is split into $3$ equal parts and one part is put-in (positive divisor). The remainder is what is remaining in $\textrm{\left(r e c e i v e d\right\rangle} 7$.

The result is quotient $- 2$ and remainder $- 1$

This is verified with $\left(- 2\right) \times 3 + \left(- 1\right) = - 7$

the division $7 \div \left(- 3\right)$ is understood as $\textrm{\left(r e c e i v e d\right\rangle} 7$ is split into $3$ equal parts and one part is taken-away (negative divisor). The remainder is what is remaining in $\textrm{\left(r e c e i v e d\right\rangle} 7$.

The result is quotient $- 2$ and remainder $1$

This is verified with $\left(- 2\right) \times \left(- 3\right) + 1 = 7$

The division $\left(- 7\right) \div \left(- 3\right)$ is understood as $\textrm{\left(g i v e n\right\rangle} 7$ is split into $3$ equal parts and one part is taken-away (negative divisor). The remainder is what is remaining in $\textrm{\left(r e c e i v e d\right\rangle} 7$.

The result is quotient $2$ and remainder $- 1$

This is verified with $2 \times \left(- 3\right) + \left(- 1\right) = - 7$

The summary of integer division illustrative examples:

•  $7 \div 2 = 3$ with $1$ remainder
$7$ received split into $2$ parts and one part is put-in = $3$ received and remainder $1$ received

•  $\left(- 7\right) \div 2 = - 3$ with $- 1$ remainder
$- 7$ given split into $2$ parts and one part is put-in = $3$ given and remainder $1$ given

•  $7 \div \left(- 2\right) = - 3$ with $1$ remainder
$7$ received split into $2$ parts and one part is taken-away = $3$ given and remainder $1$ received

•  $\left(- 7\right) \div \left(- 2\right) = 3$ with $- 1$ remainder
$7$ given split into $2$ parts and one part is taken-away = $3$ received and remainder $1$ given.

Remainder takes the sign of the dividend.

summary

Integer Division -- First Principles: Directed whole numbers division is splitting the dividend into divisor number of equal parts with direction taken into account.

If the divisor is positive, then one part is put-in.
If the divisor is negative, then one part is taken-away.
Remainder is that of the dividend retaining direction information.

Outline