### Positive and Negative Numbers (Psychology)

#### Introduction

Positive numbers are those which are greater than zero. Negative numbers are those which are less than zero. Below is a table to help you remember what to do with the sign when using multiplication or division and addition or subtraction.

\begin{array}{ccccc} \text{Positive} & \times / \div & \text{Positive} & = & \text{Positive} \\ \text{Negative} & \times / \div & \text{Negative} & = & \text{Positive} \\ \text{Positive} & \times / \div & \text{Negative} & = & \text{Negative} \\ \text{Negative} & \times / \div & \text{Positive} & = & \text{Negative} \\ \end{array}

\begin{array}{ccccc} + (-\text{Number}) &= -\text{Number} \\ - (+\text{Number}) &= -\text{Number} \\ + (+\text{Number}) &= +\text{Number} \\ - (-\text{Number}) &= +\text{Number} \\ \end{array}

For more information see positive and negative numbers in our Numeracy section.

#### Worked Examples

First we have a worked example on adding and subtracting negative numbers.

###### Worked Example

At the beginning of the year, you opened your own surgery. You have $\large\unicode{xA3}\normalsize 20,\!000$ left to spend. As with most newly established enterprises, you are making losses in the initial months. In January you made a loss of $\large\unicode{xA3}\normalsize4,\!500$, in February you lost three times as much as in January and in March you made a loss of half as much as in February. During this time you obtained three grants each of $\large\unicode{xA3}\normalsize1,\!000$.

Are you in your overdraft by the end of March? If so, by how much? If not, how much do you have left to spend?

###### Solution

Firstly calculate how much of a loss you made in February and March: In February: you lost three times as much as in January : $\large\unicode{xA3}\normalsize 4,\!500 \times 3 =\large\unicode{xA3}\normalsize 13,\!500$

' In March:' you lost half as much as in February: $\large\unicode{xA3}\normalsize13,\!500 \div 2=\large\unicode{xA3}\normalsize6,\!750$

Your losses in this period are: $\large\unicode{xA3}\normalsize4,\!500+ \large\unicode{xA3}\normalsize13,\!500+ \large\unicode{xA3}\normalsize6,\!750 = \large\unicode{xA3}\normalsize 24,\!750$. But you get three grants totalling: $\large\unicode{xA3}\normalsize1,\!000 \times 3 = \large\unicode{xA3}\normalsize3,\!000$. Hence you have: $\large\unicode{xA3}\normalsize20,\!000 + \large\unicode{xA3}\normalsize3,\!000 + (- \large\unicode{xA3}\normalsize24,\!750) = - \large\unicode{xA3}\normalsize1,\!750$ left at the end of March. So you are in your overdraft by $\large\unicode{xA3}\normalsize1,\!750$ by the end of March.

We now have a second example on multiplying/dividing by negative numbers.

###### Worked Example

The receptionist has a water dispenser on the desk to allow people waiting to have a drink while they wait.

The tank currently has $2500$ml in it. The receptionist knows that $500$ml is drunk every day.

a) How full was the tank $4$ days ago?

b) You are currently in a heat wave and twice as much water is being drunk. Will there be enough water in the dispenser for $3$ more days?

###### Solution

a)

We know that the amount of water in the dispenser changes by $-500$ml each day. To calculate the amount in the tank we subtract this amount $4$ times, (to go back $4$ days).

So the over all change is: $-4\times -500=2000$ml.

Therefore $4$ days ago there was: $2500+2000=4500$ml in the tank.

b)

The amount of water used each day has now doubled. So the overall change each day is $-1000$ml.

In $3$ days time we can calculate there will be: $2500 - 3\times 1000 = -500$ml in the tank.

Therefore there is not enough water in the dispenser to last $3$ more days as the number is negative.

#### Test Yourself

Try our Numbas tests on background mathematics.