This page extends the multiplication in first principles into a simplified procedure for multiplication of integers, which is called *sign property of integer multiplication*.

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In whole numbers, we had studied the following.**Multiplication by Place-value with Regrouping** - Simplified Procedure : Two numbers are multiplied as illustrated in the figure. Note: The procedure combines `10` units into `1` ten as carry over and so on for higher place-values.

What is the product of `4` and `3`?

By first principles, `text(received:)4` is repeatedly put in `3` times.

- `43`
- `12`
- `12`

The answer is "`12`"

What is the product of `-4` and `3`?

By first principles, `text(given:)4` is repeatedly put in `3` times.

- `-12`
- `-12`
- `+12`

The answer is "`-12`"

What is the product of `4` and `-3`?

By first principles, `text(received:)4` is repeatedly taken away `3` times.

- `-12`
- `-12`
- `+12`

The answer is "`-12`"

What is the product of `-4` and `-3`?

By first principles, `text(given:)4` is repeatedly taken away `3` times.

- `-12`
- `+12`
- `+12`

The answer is "`+12`"

Summary of integer multiplication illustrative examples:

• `4xx3 = 12`

`text(received:)4` is put-in `3` times

• `(-4)xx3 = -12`

`text(given:)4` is put-in `3` times

• `4xx(-3) = -12`

`text(received:)4` is taken-away `3` times

• `(-4)xx(-3) = 12`

`text(given:)4` is taken-away `3` times

Based on this, the multiplication is simplified as

• +ve `xx` +ve `=` +ve

• -ve `xx` +ve `=` -ve

• +ve `xx` -ve `=` -ve

• -ve `xx` -ve `=` +ve

**Integer Multiplication -- Simplified Procedure** : The sign of the product is decided by signs of multiplicand and multiplier as follows.**Sign-Property of Integer Multiplication**

• positive `xx` positive `=` positive

• positive `xx` negative `=` negative

• negative `xx` positive `=` negative

• negative `xx` negative `=` positive

The absolute value of the product is calculated by whole number multiplication of absolute values of the multiplicand and multiplier.

*Solved Exercise Problem: *

Find the product of the multiplication `22 xx (-1)`

- `-11`
- `-22`
- `-22`

The answer is "`-22`"

*Solved Exercise Problem: *

Find the product of the multiplication `0xx(-32)`

- `0`
- `-0`
- both the above
- both the above

The answer is "both the above"

*Solved Exercise Problem: *

Find the product of the multiplication `-222 xx (-88)`

- `-19536`
- `19536`
- `19536`

The answer is "a positive number".

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