- Sections:
- 0, Chapter 1, Introduction,
### Introduction

You are no doubt familiar with negative numbers. In this unit we will review the rules for manipulating such numbers.

The unit aims to give participants an understanding of

- the concept of negative numbers
- the addition and subtraction of negative numbers
- multiplication and division of negative numbers.

There are three sections in this Unit, namely:

- Negative Numbers
- Addition and Subtraction
- Multiplication and Division

- 0, Chapter 2, Negative Numbers,
### Negative Numbers

We start with a quick review of key facts:

**Positive numbers**are numbers greater than zero.**Negative numbers**are numbers less than zero.

**Zero**is neither positive nor negative.A number line illustrates positive and negative numbers. Negative numbers are always written with a

**'−'**sign in front of them. They are counted down from zero to the left on a number line.**Note**: You can extend this number line as far as you like/need in either direction. - 1, Chapter 2, Task 1, Examples of Negative Numbers,
### Examples of Negative Numbers

Here are some examples of negative numbers:

**Temperature**:

Thermometers measure both negative and positive values.

For example, the maximum temperature reached in Hull last year was

\(32\)°C, and the minimum \(- 9\)°C.**Altitude**:

Places below sea level (such as the Dead Sea in Israel, and Death

Valley in California) have negative values for altitude; places above

sea level have positive values for altitude.**Sports**:

For example, in golf ‘\(3\) below par’ is recorded as \(-3\).**Money**:

If the amount of money in a bank account goes ‘into the red’, then the

bank account holder owes the bank money (that is, they are in debt);

this amount is shown by a negative value, for example: \(-\)£\(234.25\). - 1, Chapter 2.1, Inequalities Signs,
### Inequalities Signs

We also use inequality signs with negative numbers.

The sign \(\lt\) means “is less than” as, for example \(4 \lt 7\).

The sign \(\gt\) means “is greater than” as in \(8 \gt 4\).You can see, using the number line that:

\(\begin{align*} -2 &\lt 5\\ -7 &\lt -3\\ 5 &\gt -4\\ -2 &\gt -8 \end{align*}\)

Based on the above, is this statement true or false?

**\(3 \gt -4\)**This is true; \(3\) is on the right of \(-4\) on the number line

- 1, Chapter 2.1, Task 1, Fitness Check 1,
### Exercises

**Exercise 1**Put inequality signs, \(\lt\) or \(\gt\), between each pair of numbers to give a true statement:

a. \(-3\)

a. \(-3 \lt 4\)

b. \(-6\)

b. \(-6 \gt -7\)

c. \(2\)

c. \(2 \gt -4\)

**Exercise 2**Write the set of numbers in order with the smallest first:

\(3\), \(-2\), \(8\), \(0\), \(-1\), \(1\), \(-3\)

Answer: \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(3\), \(8\)

**Exercise 3**Write down all integers that could go in box below to make the statement true:

\(-2 \lt\) ............... \(\lt 4\)

Answer: \(-2 \lt\) \(-1\), \(0\), \(1\), \(2\), \(3\) \(\lt 4\)

- 0, Chapter 3, Adding and Subtraction,
### Adding and subtracting positive and negative numbers

To add and subtract numbers always begin counting from zero.

To add a**positive number**, move to the right on a number line.

To add a**negative number**, move to the left on a number line.#### Negative Numbers: Adding

Let's calculate \(-3 + 8\)

To add -3 and 8, draw the starting number on a number line and count to the right when adding numbers. We start at \(-3\) and move \(8\) to the right so, \(-3 + 8 = 5\)

#### Negative Numbers: Substracting

What is \(5 - 7\)?

To subtract \(7\) from \(5\), start at \(5\) on a number line, then count \(7\) to the left. This will give you the answer \(-2\)

Here is another example, let's calculate \(4 - \left(-2\right)\).

We start at \(4\) and move \(2\) to the right so \(4 - \left(-2\right) = 6\)

- 1, Chapter 3, Task 1, Exercises 2,
### Exercises

**Exercise 1**Calculate

a. \(-5 + 8\)

a. \(3\)

b. \(-6 + 6\)

b. \(0\)

c. \(5 - 7\)

c. \(-2\)

d. \(5 - (-6)\)

d. \(11\)

**Exercise 2**Write done two possible calculations shown below:

Answer: \(-2 + 5 = 3\) and \(-2 - (-5) = 3\)

**Exercise 3**Copy and complete this addition table; the first row has been done.

\(+\) \(-4\) \(-2\) \(0\) \(2\) \(4\) \(-3\) \(-7\) \(-5\) \(-3\) \(-1\) \(1\) \(-1\) \(1\) \(3\)

\(+\) \(-4\) \(-2\) \(0\) \(2\) \(4\) \(-3\) \(-7\) \(-5\) \(-3\) \(-1\) \(1\) \(-1\) \(-5\) \(-3\) \(-1\) \(1\) \(3\) \(1\) \(-3\) \(-1\) \(1\) \(3\) \(5\) \(3\) \(-1\) \(1\) \(3\) \(5\) \(7\)

**Exercise 4**Overnight the temperature dropped from \(5\)°C to \(−14\)°C.

How many degrees did it fall?Answer: \(19\)°C

- 0, Chapter 4, Multiplication and Division,
### Negative Numbers: Multiplication & Division

The rule for multiplying and dividing is very similar to the rule for adding and subtracting.

- When the signs are different the answer is negative.
- When the signs are the same the answer is positive.

A

**positive number**\(\times\) a**negative number**gives a**negative**;

a**negative number**\(\times\) a**negative number**gives a**positive**;The table on the left shows what happens to the sign of the answer when positive and negative numbers are

**multiplied**; the same table can be used for**division**of positive and negative numbers.For example, \(5 \times (-7) = -35\) and \((-8) \times (-7) = 56\).

Similarly, \(18 \div (-3) = -6\) and \((-21) \div (-3) = 7\).

- 1, Chapter 4, Task 1, Exercises 3,
### Exercises

**Exercise 1**Copy and complete this multiplication grid:.

\(\times\) \(1\) \(2\) \(3\) \(4\) \(-1\) \(-2\) \(-3\) \(-4\)

\(\times\) \(1\) \(2\) \(3\) \(4\) \(-1\) \(-1\) \(-2\) \(-3\) \(-4\) \(-2\) \(-2\) \(-4\) \(-6\) \(-8\) \(-3\) \(-3\) \(-6\) \(-9\) \(-12\) \(-4\) \(-4\) \(-8\) \(-12\) \(-16\)

**Exercise 2**Calculate these multiplications and divisions:

a. \((-6) \times 3\)

a. \(-18\)

b. \((-3) \times (-8)\)

b. \(24\)

c. \((-6) \div 3\)

c. \(-2\)

d. \(8 \div (-2)\)

d. \(-4\)

**Exercise 3**Calculate \((-6 + 10) \div (-2)\)

Answer: \(-2\)

- 0, Chapter 5, Summary,
### Summary

Number line

A number line is shown below.

Note that we can use inequalities signs, \(\lt\) and \(\gt\) to write that

\(2 \gt −1\) (2 is greater than -1)

\(-3 \lt 3\) (-3 is less than 3)

\(-5 \lt 2\) (-5 is less than -2)

\(-1 \gt 4\) (-1 is greater than -4)

#### Addition and subtraction

A number line can be used to help in calculation. For example,

\(2-3=-1\)

\(-3+6=3\)

\(-2+-3=-5\)

#### Multiplication and division

Note that

\(\left(−1\right) \times \left(−1\right) = 1\)

\(\left(-1\right) \times 1 = -1\)

\(1\times \left(−1\right) = -1\)

\(1\times 1=1\)

and similarly for division; for example,\(\left(−4\right) \times 2= -8,\;\;\left(−4\right) \times \left(−2\right)= 8\)

\(4\times \left(−2\right) = -8,\;\;4\times 2=8\)

\(\left(−4\right) \div 2=-2,\;\;\left(−4\right) \div \left(−2\right)= 2\)

\(4\div \left(−2\right) = -2,\;\;4\div2=2\)

- 0, Chapter 1, Introduction,
- Interactive Exercises:
- Negative Numbers Interactive Exercises, https://www.cimt.org.uk/sif/number/n6/interactive.htm
- Negative Numbers, https://www.cimt.org.uk/sif/number/n6/interactive/s1.html
- Addition and Substraction, https://www.cimt.org.uk/sif/number/n6/interactive/s2.html
- Multiplication and Division, https://www.cimt.org.uk/sif/number/n6/interactive/s3.html

- YouTube URL: https://youtu.be/osu0a5vRokA
- External Links:
- Negative Numbers Quiz, https://www.bbc.co.uk/bitesize/articles/zjbk8xs, images/Images/bitesize-logo.png, For more on negative numbers and a fun quiz try this link to BBC bitesize
- Multiplying Positive and Negative Numbers, https://youtu.be/QmarTb7wXro, images/Images/math-club.png, Think your family is weird? This family's eccentricities get multiplied, like, tenfold.

- File Attachments: media/course-resources/Negative-Numbers_Essential-Information.pdfmedia/course-resources/Negative-Numbers_Learning-Objectives.pdfmedia/course-resources/Negative-Numbers_PowerPoint-Presentation.pptxmedia/course-resources/Negative-Numbers_Text.pdfmedia/course-resources/Negative-Numbers_Answers.pdf