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mathsIntegersInteger Multiplication & Division

Integer Division : First Principles

The first principles of division is, splitting a quantity into a number of parts and count or measure one part. This page explains the same for integers, which are directed whole numbers.



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In whole numbers, `6-:2` means: dividend `6` is split into `2` equal parts and one part is put in.

In integers, `2` and `-2` are understood as
`text(received:)2=2` and `text(given:2)=-2`.

It is also called `text(aligned:)2=2` and `text(opposed: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.

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 `text(received:)6=6` and `text(received:)2=2`.

The number of candies received is
`text(received:)6=6` is split into `2` parts and one part `text(received:)3=3` is put-in
`6-:2=3`

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 `text(given:)6=-6` and `text(received:)2=2`. The number of candies received is
`text(given:)6=-6` is split into `2` parts and one part `text(given:)3=-3` is put-in
`(-6)-:2=-3`

Considering division of `(-6)-:2`. The numbers are given in integer form. The numbers in directed whole numbers form are `text(given:)6` and `text(received:)2`.

The Division is explained as
`text(given:)6=-6` is the dividend
`text(received:)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 `=text(given:)3`.

The same in integer form
`=(-6)-:2`
`=-3`

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 `text(received:)6=6` and `text(given:)2=-2`.

The number of candies received is
`text(received:)6=6` is split into `2` parts and one part `text(received:)3=3` is taken away, which is `text(given:)3=-3`
`6-:(-2)=-3`

Considering division of `6-:(-2)`. The numbers are given in integer form. To understand first principles of division, let us convert that to directed whole numbers form `text(received:)6` and `text(given:)2`.

The Division is explained as
`text(received:)6=6` is the dividend
`text(given:)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. `text(received:)3` taken away is `text(given:)3`.
Thus the quotient of the division is `=text(given:)3`.

The same in integer form
`=6-:(-2)`
`=-3`

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 `text(given:)6=-6` and `text(given:)2=-2`.

The number of candies received is
`text(given:)6=-6` is split into `2` parts and one part `text(given:)3=-3` is taken-away, which is `text(received:)3=+3`
`(-6)-:(-2)=+3`

Considering division of `(-6)-:(-2)`. The numbers are given in integer form. To understand first principles of division, let us convert that to directed whole numbers form `text(given:)6` and `text(given:)2`.

The Division is explained as
`text(given:)6=-6` is the dividend
`text(given:)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. `text(given:)3` taken away is `text(received:)3`.
Thus the quotient of the division is `=text(received:)3`.

The same in integer form
`=(-6)-:(-2)`
`=3`

The summary of integer division illustrative examples:

 •  `6-:2 = 3`
`6` received split into `2` parts and one part is put-in = `3` received

 •  `(-6)-:2 = -3`
`6` given split into `2` parts and one part is put-in = `3` given

 •  `6-:(-2) = -3`
`6` received split into `2` parts and one part is taken-away = `3` given

 •  `(-6)-:(-2) = 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.

The division `7-:3` is understood as `text(received:)7` is split into `3` equal parts and one part is put-in (positive divisor). The remainder is what is remaining in `text(received:)7`.

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

This is verified with `2xx3 + 1 = 7` (quotient multiplied divisor + remainder = dividend )

The division `(-7)-:3` is understood as `text(given:)7` is split into `3` equal parts and one part is put-in (positive divisor). The remainder is what is remaining in `text(received:)7`.

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

This is verified with `(-2)xx3 + (-1) = -7`

the division `7-:(-3)` is understood as `text(received:)7` is split into `3` equal parts and one part is taken-away (negative divisor). The remainder is what is remaining in `text(received:)7`.

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

This is verified with `(-2)xx(-3) + 1 = 7`

The division `(-7)-:(-3)` is understood as `text(given:)7` is split into `3` equal parts and one part is taken-away (negative divisor). The remainder is what is remaining in `text(received:)7`.

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

This is verified with `2xx(-3) + (-1) = -7`

The summary of integer division illustrative examples:

 •  `7-:2 = 3` with `1` remainder
`7` received split into `2` parts and one part is put-in = `3` received and remainder `1` received

 •  `(-7)-:2 = -3` with `-1` remainder
`-7` given split into `2` parts and one part is put-in = `3` given and remainder `1` given

 •  `7-:(-2) = -3` with `1` remainder
`7` received split into `2` parts and one part is taken-away = `3` given and remainder `1` received

 •  `(-7)-:(-2) = 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.

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.

                            
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