maths > integers

Integer Subtraction : First Principles

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overview

In whole numbers, we studied that, Subtraction in first principles is taking away a part of the quantity and counting or measuring the remaining quantity.

Directed whole numbers subtraction is taking away an amount from another. This is handled as inverse of addition with direction information taken into account.

The subtraction for integers is examined for all combinations of positive and negative integers as minuend and subtrahend.

take away

In whole numbers, we had studied the following.

Subtraction - First Principles : Two numbers are considered, each of which represents a count or measurement. From one amount represented by the first number, the amount represented by the second is taken away to form a result representing the remaining amount. The count or measurement of the remaining amount is the result of subtraction.

eg: $20-13=7$$20 - 13 = 7$

$20$$20$ is the minuend

$13$$13$ is the subtrahend

$7$$7$ is the difference

$5-2=3$$5 - 2 = 3$.

This can be explained as either one of the following.
•  $2$$2$ is given from $5$$5$
•  $2$$2$ is taken-away from $5$$5$

In the context of integers with $\text{received:}$$\textrm{\left(r e c e i v e d\right\rangle}$ and $\text{given:}$$\textrm{\left(g i v e n\right\rangle}$, subtraction is referred as taken-away.

A girl has a box of candies. The number of candies in the box is not counted. But she keeps track of how many candies she receives or how many she gives away. She maintains a daily account of how many are received or given.

She made two transactions, $\text{received:}5=5$$\textrm{\left(r e c e i v e d\right\rangle} 5 = 5$ candies and then take-away $\text{received:}2=2$$\textrm{\left(r e c e i v e d\right\rangle} 2 = 2$ candies. The remaining candies with her is $5-2=3$$5 - 2 = 3$

take away a given

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

She made two transactions, $\text{received:}5=5$$\textrm{\left(r e c e i v e d\right\rangle} 5 = 5$ candies and then taken-away $\text{given:}2=-2$$\textrm{\left(g i v e n\right\rangle} 2 = - 2$ candies.

The remaining candies with her is $5-\left(-2\right)=7$$5 - \left(- 2\right) = 7$

Considering subtraction of $-2$$- 2$ from $5$$5$. The numbers are given in integer form. To understand first principles of subtraction, let us convert that to directed whole numbers form $\text{received:}5$$\textrm{\left(r e c e i v e d\right\rangle} 5$ and $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$.

The subtraction is explained as
$\text{received:}5$$\textrm{\left(r e c e i v e d\right\rangle} 5$ is the minuend
$\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$ is taken-away from the minuend
taking away $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$ is effectively $\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$
$=\text{received:}7$$= \textrm{\left(r e c e i v e d\right\rangle} 7$ is the result.

The same in integer form
$=5-\left(-2\right)$$= 5 - \left(- 2\right)$
$=5+2$$= 5 + 2$
$=7$$= 7$

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

She made two transactions, $\text{given:}5=-5$$\textrm{\left(g i v e n\right\rangle} 5 = - 5$ candies and then taken-away $\text{received:}2=2$$\textrm{\left(r e c e i v e d\right\rangle} 2 = 2$ candies.

So, the remaining candies with her is $\left(-5\right)-2=-7$$\left(- 5\right) - 2 = - 7$

Considering subtraction of $2$$2$ from $-5$$- 5$. The numbers are given in integer form. To understand first principles of subtraction, let us convert that to directed whole numbers form $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ and $\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$.

The subtraction is explained as
$\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ is the minuend
$\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$ is taken-away from the minuend
$=\text{given:}7$$= \textrm{\left(g i v e n\right\rangle} 7$ is the result.

The same in integer form
$=-5-2$$= - 5 - 2$
$=\left(-5\right)+\left(-2\right)$$= \left(- 5\right) + \left(- 2\right)$
$=-7$$= - 7$

take away smaller given

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

She made two transactions, $\text{given:}5=-5$$\textrm{\left(g i v e n\right\rangle} 5 = - 5$ candies and then taken-away $\text{given:}2=-2$$\textrm{\left(g i v e n\right\rangle} 2 = - 2$ candies.

So, The remaining candies with her is $-5-\left(-2\right)=-3$$- 5 - \left(- 2\right) = - 3$.

Considering subtraction of $-2$$- 2$ from $-5$$- 5$. The numbers are given in integer form. The number in directed whole numbers form are $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ and $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$.

The subtraction is explained as
$\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ is the minuend
$\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$ is taken-away from the minuend
taking away $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$ is effectively $\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$.
$=\text{given:}3$$= \textrm{\left(g i v e n\right\rangle} 3$ is the result.

The same in integer form
$=-5-\left(-2\right)$$= - 5 - \left(- 2\right)$
$=-5+2$$= - 5 + 2$
$=-3$$= - 3$

take away larger given

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

She made two transactions, $\text{given:}5=-5$$\textrm{\left(g i v e n\right\rangle} 5 = - 5$ candies and then taken-away $\text{given:}7=-7$$\textrm{\left(g i v e n\right\rangle} 7 = - 7$ candies.

So, The remaining candies with her is $-5-\left(-7\right)=-5+7=+2$$- 5 - \left(- 7\right) = - 5 + 7 = + 2$

Considering subtraction of $-7$$- 7$ from $-5$$- 5$. The number are given in integer form. The numbers in directed whole numbers form are $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ and $\text{given:}7$$\textrm{\left(g i v e n\right\rangle} 7$.

The subtraction is explained as
$\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ is the minuend
$\text{given:}7$$\textrm{\left(g i v e n\right\rangle} 7$ is taken-away from the minuend
taking away $\text{given:}7$$\textrm{\left(g i v e n\right\rangle} 7$ is effectively $\text{received:}7$$\textrm{\left(r e c e i v e d\right\rangle} 7$.
$=\text{received:}2$$= \textrm{\left(r e c e i v e d\right\rangle} 2$ is the result.

The same in integer form
$=-5-\left(-7\right)$$= - 5 - \left(- 7\right)$
$=-5+7$$= - 5 + 7$
$=2$$= 2$

The summary of integer subtraction illustrative examples:

•  $5-2=5+\left(-2\right)=3$$5 - 2 = 5 + \left(- 2\right) = 3$
$\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$ taken away from $\text{received:}5$$\textrm{\left(r e c e i v e d\right\rangle} 5$ results in combining $\text{received:}5$$\textrm{\left(r e c e i v e d\right\rangle} 5$ and $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$

•  $5-\left(-2\right)=5+\left(+2\right)=7$$5 - \left(- 2\right) = 5 + \left(+ 2\right) = 7$
$\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$ taken away from $\text{received:}5$$\textrm{\left(r e c e i v e d\right\rangle} 5$ results in combining $\text{received:}5$$\textrm{\left(r e c e i v e d\right\rangle} 5$ and $\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$

•  $-5-2=-5+\left(-2\right)=-7$$- 5 - 2 = - 5 + \left(- 2\right) = - 7$
$\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$ taken away from $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ results in combining $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ and $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$

•  $-5-\left(-2\right)=-5+\left(+2\right)=-3$$- 5 - \left(- 2\right) = - 5 + \left(+ 2\right) = - 3$
$\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$ taken away from $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ results in combining $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ and $\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$

•  $-5-\left(-7\right)=-5+\left(+7\right)=2$$- 5 - \left(- 7\right) = - 5 + \left(+ 7\right) = 2$
$\text{given:}7$$\textrm{\left(g i v e n\right\rangle} 7$ taken away from $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ results in combining $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ and $\text{received:}7$$\textrm{\left(r e c e i v e d\right\rangle} 7$

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

The same is captured in subtraction is handled as inverse of addition.

summary

Integer Subtraction -- First Principles : Directed whole numbers subtraction is taking away an amount from another. This is handled as inverse of addition with direction information taken into account.

Examples are :
•  from $\text{received:}5$$\textrm{\left(r e c e i v e d\right\rangle} 5$, taking away $\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$ is equivalently, combining $\text{received:}5$$\textrm{\left(r e c e i v e d\right\rangle} 5$ and $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$;
$5-2=5+\left(-2\right)$$5 - 2 = 5 + \left(- 2\right)$
•  from $\text{received:}5$$\textrm{\left(r e c e i v e d\right\rangle} 5$, taking away $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$ is equivalently, combining $\text{received:}5$$\textrm{\left(r e c e i v e d\right\rangle} 5$ and $\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$;
$5-\left(-2\right)=5+\left(+2\right)$$5 - \left(- 2\right) = 5 + \left(+ 2\right)$
•  from $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$, taking away $\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$ is equivalently, combining $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ and $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$;
$\left(-5\right)-2=\left(-5\right)+\left(-2\right)$$\left(- 5\right) - 2 = \left(- 5\right) + \left(- 2\right)$
•  from $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$, taking away $\text{given:}2$$\textrm{\left(g i v e n\right\rangle} 2$ is equivalently, combining $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ and $\text{received:}2$$\textrm{\left(r e c e i v e d\right\rangle} 2$;
$\left(-5\right)-\left(-2\right)=\left(-5\right)+\left(+2\right)$$\left(- 5\right) - \left(- 2\right) = \left(- 5\right) + \left(+ 2\right)$
•  from $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$, taking away $\text{given:}7$$\textrm{\left(g i v e n\right\rangle} 7$ is equivalently, combining $\text{given:}5$$\textrm{\left(g i v e n\right\rangle} 5$ and $\text{received:}7$$\textrm{\left(r e c e i v e d\right\rangle} 7$;
$\left(-5\right)-\left(-7\right)=\left(-5\right)+\left(+7\right)$$\left(- 5\right) - \left(- 7\right) = \left(- 5\right) + \left(+ 7\right)$

Outline