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, what does `6-:2` mean?

- division is just division, no meaning
- dividend `6` is split into `2` equal parts and one part is put in
- dividend `6` is split into `2` equal parts and one part is put in

The answer is "dividend `6` is split into `2` equal parts and one part is put in"

In integers what do `2` and `-2` mean?

- no meaning for the numbers
- `text(received:)2=2` and `text(given:2)=-2`
- `text(received:)2=2` and `text(given:2)=-2`

The answer is "`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, what do a positive or a negative divisor represent?

- positive divisor represents: one part is put-in
- negative divisor represents: one part is taken-away
- both the above
- both the above

The answer is "both the above". 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`. How many candies are received?

- `6-:2=3`
- `text(received:)6=6` is split into `2` parts and one part `text(received:)3=3` is put-in
- both the above
- both the above

The answer is "both the above".

*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`. How many candies are received?

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

The answer is "`-3`". This is explained in the next page.

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`. How many candies are received?

- `6-:2=+3`
- `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`
- `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`

The answer is "`-3`". This is explained in the next page.

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`. How many candies are received?

- `(-6)-:(-2)=-3`
- `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`
- `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`

The answer is "`3`". This is explained in the next page.

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.

What is the result of the division `7-:3`?

By first principles, `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`.

- `3`
- quotient `2` and remainder `1`
- quotient `2` and remainder `1`

The answer is "quotient `2` and remainder `1`".

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

What is the result of the division `(-7)-:3`?

By first principles, `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`.

- quotient `-2` and remainder `+1`
- quotient `-2` and remainder `-1`
- quotient `-2` and remainder `-1`

The answer is "quotient `-2` and remainder `-1`".

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

What is the result of the division `7-:(-3)`?

By first principles, `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`.

- quotient `-2` and remainder `+1`
- quotient `-2` and remainder `+1`
- quotient `-2` and remainder `-1`

The answer is "quotient `-2` and remainder `1`".

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

What is the result of the division `(-7)-:(-3)`?

By first principles, `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`.

- quotient `2` and remainder `+1`
- quotient `2` and remainder `-1`
- quotient `2` and remainder `-1`

The answer 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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