maths > integers

Integer Multiplication : First Principles

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overview

This page introduces multiplication of integers as -- the multiplicand, is repeated the number of times given by the multiplier. The combined result of the repetition is counted or measured as the product of the multiplication.

The multiplication when multiplicand, multiplier are positive or negative integers is discussed.

The definition of multiplication in first principles form the basis to understanding the simplified procedure for multiplication of large integers.

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: $12×3=36$$12 \times 3 = 36$

$12$$12$ is the multiplicand

$3$$3$ is the multiplier

$36$$36$ is the product

$3×2$$3 \times 2$ means multiplicand $3$$3$ is repeatedly put-in multiplier $2$$2$ times.

In integers, $2$$2$ and $-2$$- 2$ mean,

$\text{received:}2=2$$\textrm{\left(r e c e i v e d\right\rangle} 2 = 2$ and $\text{given:2}=-2$$\textrm{g i v e n : 2} = - 2$

It is also referred as \text{aligned:}2=2$\textrm{\left(a l i g \ne d\right\rangle} 2 = 2$ and $\text{opposed:2}=-2$$\textrm{o p p o s e d : 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$$2$ received is put in $3$$3$ times. The numbers in the integer forms are $\text{received:}2=2$$\textrm{\left(r e c e i v e d\right\rangle} 2 = 2$ and $\text{received:}3=3$$\textrm{\left(r e c e i v e d\right\rangle} 3 = 3$.

$2×3=6$$2 \times 3 = 6$ OR
$\text{received:}2×\phantom{\rule{1ex}{0ex}}\text{received:}3=\text{received:}6$$\textrm{\left(r e c e i v e d\right\rangle} 2 \times \textrm{\left(r e c e i v e d\right\rangle} 3 = \textrm{\left(r e c e i v e d\right\rangle} 6$

given is repeated

In the box, $2$$2$ given is put in $3$$3$ times. The numbers in the integer forms are $\text{given:}2=-2$$\textrm{\left(g i v e n\right\rangle} 2 = - 2$ and $\text{received:}3=3$$\textrm{\left(r e c e i v e d\right\rangle} 3 = 3$.

The number of candies received is
$\text{given:}2×\text{received:}3=\text{given:}6=-6$$\textrm{\left(g i v e n\right\rangle} 2 \times \textrm{\left(r e c e i v e d\right\rangle} 3 = \textrm{\left(g i v e n\right\rangle} 6 = - 6$.

Considering multiplication of $-2$$- 2$ and $3$$3$. The numbers are in integer form. The numbers in directed whole numbers form are $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$ and $\text{received:}3$$\textrm{\left(r e c e i v e d\right\rangle} 3$.

The multiplication is explained as
$\text{given:}2=-2$$\textrm{\left(g i v e n\right\rangle} 2 = - 2$ is the multiplicand
$\text{received:}3$$\textrm{\left(r e c e i v e d\right\rangle} 3$ is multiplier

Multiplication is multiplicand repeatedly put-in multiplier number of times.
$-2$$- 2$ put-in $3$$3$ times is given as $-2+\left(-2\right)+\left(-2\right)=-6$$- 2 + \left(- 2\right) + \left(- 2\right) = - 6$.
Thus the product of the multiplication is $=\text{given:}6$$= \textrm{\left(g i v e n\right\rangle} 6$.

The same in integer form
$=\left(-2\right)×3$$= \left(- 2\right) \times 3$
$=-6$$= - 6$

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

In the box, $2$$2$ received is taken-away $3$$3$ times. The numbers in the integer forms are $\text{received:}2=2$$\textrm{\left(r e c e i v e d\right\rangle} 2 = 2$ and $\text{given:}3=-3$$\textrm{\left(g i v e n\right\rangle} 3 = - 3$ (equivalent of multiplier taken-away).

The number of candies received is
$\text{received:}2×\text{given:}3=\text{given:}6=-6$$\textrm{\left(r e c e i v e d\right\rangle} 2 \times \textrm{\left(g i v e n\right\rangle} 3 = \textrm{\left(g i v e n\right\rangle} 6 = - 6$

Considering multiplication of $2$$2$ and $-3$$- 3$. The numbers are given in integer form. The numbers in directed whole numbers form are $\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$ and $\text{given:}3$$\textrm{\left(g i v e n\right\rangle} 3$.

The multiplication is explained as
$\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$ is the multiplicand
$\text{given:}3=-3$$\textrm{\left(g i v e n\right\rangle} 3 = - 3$ is multiplier

Multiplication is multiplicand repeatedly put-in multiplier number of times.
Putting in $\text{given:}3$$\textrm{\left(g i v e n\right\rangle} 3$ times is equivalently taking away $\text{received:}3$$\textrm{\left(r e c e i v e d\right\rangle} 3$ times. This was explained in lesson handling signs

$2$$2$ taken away $3$$3$ times is given as $-2-2-2=-6$$- 2 - 2 - 2 = - 6$.
Thus the product of the multiplication is $=\text{given:}6$$= \textrm{\left(g i v e n\right\rangle} 6$.

The same in integer form
$=2×\left(-3\right)$$= 2 \times \left(- 3\right)$
$=-6$$= - 6$

given is repeatedly taken away

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

In the box, $2$$2$ given is taken away $3$$3$ times. The numbers in the integer forms are $\text{given:}2=-2$$\textrm{\left(g i v e n\right\rangle} 2 = - 2$ and $\text{given:}3=-3$$\textrm{\left(g i v e n\right\rangle} 3 = - 3$ (equivalent of multiplier taken-away).

The number of candies received is
$\text{given:}2×\text{given:}3=\text{received:}6=6$$\textrm{\left(g i v e n\right\rangle} 2 \times \textrm{\left(g i v e n\right\rangle} 3 = \textrm{\left(r e c e i v e d\right\rangle} 6 = 6$.

Considering multiplication of $-2$$- 2$ and $-3$$- 3$. The numbers are given in integer form. The numbers in directed whole numbers form are $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$ and $\text{given:}3$$\textrm{\left(g i v e n\right\rangle} 3$.

The multiplication is explained as
$\text{given:}2=-2$$\textrm{\left(g i v e n\right\rangle} 2 = - 2$ is the multiplicand
$\text{given:}3=-3$$\textrm{\left(g i v e n\right\rangle} 3 = - 3$ is the multiplier

Multiplication is multiplicand repeatedly put-in multiplier number of times.
Putting in $\text{given:}3$$\textrm{\left(g i v e n\right\rangle} 3$ times is equivalently taking away $\text{received:}3$$\textrm{\left(r e c e i v e d\right\rangle} 3$ times. This was explained in lesson handling signs

$-2$$- 2$ put-in $-3$$- 3$ times is given as $-\left(-2\right)-\left(-2\right)-\left(-2\right)=6$$- \left(- 2\right) - \left(- 2\right) - \left(- 2\right) = 6$.
Thus the product of the multiplication is $=\text{received:}6$$= \textrm{\left(r e c e i v e d\right\rangle} 6$.

The same in integer form
$=\left(-2\right)×\left(-3\right)$$= \left(- 2\right) \times \left(- 3\right)$
$=6$$= 6$

The summary of integer multiplication illustrative examples:

•  $2×3=6$$2 \times 3 = 6$
$2$$2$ received is put in $3$$3$ times $=6$$= 6$ received

•  $\left(-2\right)×3=-6$$\left(- 2\right) \times 3 = - 6$
$2$$2$ given is put in $3$$3$ times $=6$$= 6$ given

•  $2×\left(-3\right)=-6$$2 \times \left(- 3\right) = - 6$
$2$$2$ received is taken-away $3$$3$ times $=6$$= 6$ given

•  $\left(-2\right)×\left(-3\right)=6$$\left(- 2\right) \times \left(- 3\right) = 6$
$2$$2$ given is taken-away $3$$3$ times $=6$$= 6$ received

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

summary

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

Outline