Negative numbers

​​​​In a nutshell

Negative numbers are numbers below zero. They have a minus sign in front of them in order to show that they are negative.

Negative numbers on the number line

Negative numbers count from zero in the opposite direction to positive numbers, as shown by this number line. ​

​$$\begin{array}{cccccccccccccc}&&&&&&&&&&\leftarrow\text{negative|positive}\rightarrow\\&&&&&&&&&&\text{increasing}\\&&&&&&&&&&\longrightarrow\\-5&&-4&&-3&&-2&&-1&&\bold0&&1&&2&&3&&4&&5&&6& \\|&&|&&|&&|&&|&&|&&|&&|&&|&&|&&|&&|\\ \hline\\&&&&&&&&&&\text{decreasing}\\&&&&&&&&&&\longleftarrow\\\end{array}$$​​

​Just like with positive numbers, negative numbers get bigger in this direction: ($$\rightarrow$$​) and get smaller in this direction: ($$\leftarrow$$​​).

The closer a negative number is to zero, the larger it is.

Example 1

Which of the following numbers is the largest: $$-3, -9, -12?$$​​

$$\underline{-3}$$​ is the number that is closest to zero on the number line. Therefore it is the biggest.

Calculating intervals between positive and negative numbers

Using a number line, it is really simple to work out the interval between two numbers, regardless of whether they are positive or negative.

PROCEDURE

Note: Working out the 'interval between' numbers is the same as working out the 'difference between' them.

Calculating intervals using jumps of one

Example 2

What is the difference between $$-18$$ and $$-23$$?

​

Draw a number line starting at $$-23$$ and ending at $$-18$$.

$$\begin{array}{cccccccccccccc}-23&&-22&&-21&&-20&&-19&&-18\ \\|&&|&&|&&|&&|&&|\\ \hline\end{array}$$​​

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Count the number of jumps between the two numbers.

$$\begin{array}{cccccccccccccc}&+1&&+1&&+1&&+1&&+1&\\& \curvearrowright&&\curvearrowright &&\curvearrowright &&\curvearrowright &&\curvearrowright\\-23&&-22&&-21&&-20&&-19&&-18\ \\|&&|&&|&&|&&|&&|\\ \hline\end{array}$$​​

There are five jumps of one. 

 The difference between $$-23$$ and $$-18$$ is $$\underline{5}$$.

​

Calculating intervals using larger jumps

To work out the interval between numbers that are more spread out, you may need to use larger jumps.

Example 3

Calculate the interval between $$-15$$ and $$8$$.

Draw a number line starting at $$-15$$ and ending at $$8$$.

$$\begin{array}{cccccccccccccc}-15&&&&&-10&&&&&-5&&&&&0&&&&&5&&&8\ \\|&_|&_|&_|&_|&|&_|&_|&_|&_|&|&_|&_|&_|&_|&|&_|&_|&_|&_|&|&_|&_|&_|&_|&|\\ \hline\end{array}$$​​

Count the number of jumps between the numbers. Here it might be a good idea to go up in fives.

$$\begin{array}{cccccccccccccc}&&&+5&&&&&+5&&&&&+5&&&&&+5&&\space&+1&+1&+1\\& &&\curvearrowright &&&&&\curvearrowright &&&&&\curvearrowright&&&&&\curvearrowright&&&\curvearrowright&\curvearrowright&\curvearrowright\\-15&&&&&-10&&&&&-5&&&&&0&&&&&5&&&8\ \\|&_|&_|&_|&_|&|&_|&_|&_|&_|&|&_|&_|&_|&_|&|&_|&_|&_|&_|&|&_|&_|&_|&_|&|\\ \hline\end{array}$$​​

There are four jumps of five and three jumps of one.

​$$(5\times4)+(1\times3)=23$$

The difference between $$-15$$ and $$8$$ is $$\underline{23}$$.

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