nubtrek

Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsIntegersInteger Multiplication & Division

Integer Multiplication : First Principles

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.



click on the content to continue..

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

`3xx2` means multiplicand `3` is repeatedly put-in multiplier `2` times.

In integers, `2` and `-2` mean,

`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,

 •  positive multiplier represents repeatedly putting-in

 •  negative multiplier represents repeatedly taking-away

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`.

The total candies received is
`2xx3=6` OR
`text(received:)2 xx text( received:)3 = text(received:)6`

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`.

The number of candies received is
`text(given:)2 xx text(received:)3 = text(given:)6 = -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).

The number of candies received is
`text(received:)2 xx text(given:)3 = text(given:)6 = -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).

The number of candies received is
`text(given:)2 xx text(given:)3 = text(received:)6 = 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

                            
switch to interactive version