maths > integers

Handling Direction in Integers

what you'll learn...

overview

In the earlier page, integers were introduced as directed numbers with positive values representing received direction and negative values representing given direction. In this page, the following are explained with an example.

•  +ve of a +ve = +ve

•  -ve of a +ve = -ve

•  +ve of a -ve = -ve

•  -ve of a -ve = +ve

To explain the concepts the following mapping is used.

received:3=+3=3$\textrm{\left(r e c e i v e d\right\rangle} 3 = + 3 = 3$
given:3=3$\textrm{\left(g i v e n\right\rangle} 3 = - 3$

This models the directed numbers of various forms.
eg: temperature 3${3}^{\circ}$ above and below 0${0}^{\circ}$ is modeled as received:3=3$\textrm{\left(r e c e i v e d\right\rangle} 3 = 3$ and $\textrm{\left(g i v e n\right\rangle} 3 = - 3$.

The explanation in this is going to be little involved. Students can spend a little extra time thinking about each paragraph.

Let us consider

$\textrm{\left(r e c e i v e d\right\rangle} \left(- 3\right)$. We know that $- 3$ is $\textrm{\left(g i v e n\right\rangle} 3$ and so the expression is simplified.

$\textrm{\left(r e c e i v e d\right\rangle} \left(- 3\right) = \textrm{\left(g i v e n\right\rangle} 3 = - 3$

given a negative

Let us consider

$\textrm{\left(g i v e n\right\rangle} \left(- 3\right)$. We know that $- 3$ is $\textrm{\left(g i v e n\right\rangle} 3$ and so the expression is simplified.

$\textrm{\left(g i v e n\right\rangle} \left(- 3\right)$$= \textrm{\left(g i v e n\right\rangle} \left(\textrm{\left(g i v e n\right\rangle} 3\right)$$= \textrm{\left(r e c e i v e d\right\rangle} 3$

To understand the two numbers, $\textrm{\left(r e c e i v e d\right\rangle} \left(- 3\right)$ and $\textrm{\left(g i v e n\right\rangle} \left(- 3\right)$, let us consider an example application.

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.

She gave $3$ candies today. The number of candies received today is $- 3$.

Her brother took $3$ candies and so, she noted received or put-in $- 3$ candies today. This is represented as$\textrm{\left(r e c e i v e d\right\rangle} \left(- 3\right) = - 3$

Her brother returned $3$ candies which was given earlier and so, she noted given or taken away negative $3$ of an earlier date. Effectively this is $\textrm{\left(g i v e n\right\rangle} \left(- 3\right) = \textrm{\left(r e c e i v e d\right\rangle} 3 = + 3$

examples

What is $- \left(+ 7\right)$?
The answer is '$- 7$'.

What is $- \left(- 7\right)$?
The answer is '$7$'.

summary

received $3$
$\textrm{\left(r e c e i v e d\right\rangle} 3 = 3$
given $3$
$\textrm{\left(g i v e n\right\rangle} 3 = - 3$
received $- 3$ is put-in given 3
$\textrm{\left(r e c e i v e d\right\rangle} \left(- 3\right) = + \left(- 3\right) = - 3$
given $- 3$ is taken-away given 3
$\textrm{\left(g i v e n\right\rangle} \left(- 3\right) = - \left(- 3\right) = 3$