The first principles of multiplication is, repeatedly combine a quantity a number of times and count or measure the combined quantity. This page explains the same for integers, which are directed whole numbers.

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In whole numbers, we had studied the following.**Multiplication - First Principles** : Two numbers are considered, each of which represents a count or measurement. One of the amount is repeated the number of times given by the second amount to form the result representing the collective amount from such repetition.

eg: `12 xx 3 = 36`

`12` is the multiplicand

`3` is the multiplier

`36` is the product

In whole numbers, what does `3xx2` mean?

- multiplication is just multiplication. No meaning
- multiplicand `3` is repeatedly put-in multiplier `2` times
- multiplicand `3` is repeatedly put-in multiplier `2` times

The answer is "multiplicand `3` is repeatedly put-in multiplier `2` times"

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 referred as `text(aligned:)2=2` and `text(opposed:2)=-2`

Integers are "directed" whole numbers. A whole-number multiplication represents repeating the multiplicand multiplier number of times.

In integers, What do a positive or a negative multiplier represent?

- positive multiplier represents repeatedly putting-in
- negative multiplier represents repeatedly taking-away
- both the above
- both the above

The answer is "both the above".

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.

In the box, `2` received is put in `3` times. The numbers in the integer forms are `text(received:)2=2` and `text(received:)3=3`.

How many candies are received?

- `2xx3=6`
- `text(received:)2 xx text( received:)3 = text(received:)6`
- 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.

In the box, `2` given is put in `3` times. The numbers in the integer forms are `text(given:)2=-2` and `text(received:)3=3`.

How many candies are received?

- `2xx3=6`
- `text(given:)2 xx text(received:)3 = text(given:)6 = -6`
- `text(given:)2 xx text(received:)3 = text(given:)6 = -6`

The answer is "`-6`". This is explained in the next page.

Considering multiplication of `-2` and `3`. The numbers are given in integer form. The numbers in directed whole numbers form are `text(given:)2` and `text(received:)3`.

The multiplication is explained as

`text(given:)2=-2` is the multiplicand

`text(received:)3` is multiplier

Multiplication is multiplicand repeatedly put-in multiplier number of times.

`-2` put-in `3` times is given as `-2+(-2)+(-2) = -6`.

Thus the product of the multiplication is `=text(given:)6`.

The same in integer form

`=(-2)xx3`

`=-6`

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

In the box, `2` received is taken-away `3` times. The numbers in the integer forms are `text(received:)2=2` and `text(given:)3=-3` (equivalent of multiplier taken-away).

How many candies are received?

- `2xx3=6`
- `text(received:)2 xx text(given:)3 = text(given:)6 = -6`
- `text(received:)2 xx text(given:)3 = text(given:)6 = -6`

The answer is "`-6`". This is explained in the next page.

Considering multiplication of `2` and `-3`. The numbers are given in integer form. The numbers in directed whole numbers form are `text(received:)2` and `text(given:)3`.

The multiplication is explained as

`text(received:)2` is the multiplicand

`text(given:)3 = -3` is multiplier

Multiplication is multiplicand repeatedly put-in multiplier number of times.

Putting in `text(given:)3` times is equivalently taking away `text(received:)3` times. This was explained in lesson *handling signs*

`2` taken away `3` times is given as `-2-2-2 = -6`.

Thus the product of the multiplication is `=text(given:)6`.

The same in integer form

`=2xx(-3)`

`=-6`

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

In the box, `2` given is taken away `3` times. The numbers in the integer forms are `text(given:)2=-2` and `text(given:)3=-3` (equivalent of multiplier taken-away).

How many candies are received?

- two negative numbers cannot be multiplied
- `text(given:)2 xx text(given:)3 = text(received:)6 = 6`
- `text(given:)2 xx text(given:)3 = text(received:)6 = 6`

The answer is "`6`". This is explained in the next page.

Considering multiplication of `-2` and `-3`. The numbers are given in integer form. The numbers in directed whole numbers form are `text(given:)2` and `text(given:)3`.

The multiplication is explained as

`text(given:)2=-2` is the multiplicand

`text(given:)3=-3` is the multiplier

Multiplication is multiplicand repeatedly put-in multiplier number of times.

Putting in `text(given:)3` times is equivalently taking away `text(received:)3` times. This was explained in lesson *handling signs*

`-2` put-in `-3` times is given as `-(-2)-(-2)-(-2) = 6`.

Thus the product of the multiplication is `=text(received:)6`.

The same in integer form

`=(-2)xx(-3)`

`=6`

The summary of integer multiplication illustrative examples:

• `2xx3 = 6`

`2` received is put in `3` times `= 6` received

• `(-2)xx3 = -6`

`2` given is put in `3` times `= 6` given

• `2xx(-3) = -6`

`2` received is taken-away `3` times `= 6` given

• `(-2)xx(-3) = 6`

`2` given is taken-away `3` times `= 6` received

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

**Integer Multiplication First Principles** : Directed whole numbers multiplication is repeating the multiplicand the multiplier number of times with direction taken into account.

Repeating positive number of times is represented as repeatedly putting-in

Repeating negative number of times is represented as repeatedly taking-away

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