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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, 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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