Negative Numbers



Introduction

Negative numbers are a way of expressing values below zero

As good everyday examples, consider the case of temperatures or bank accounts.


example

If you are given a problem like

- 4 - 6 = ?

think of what happens with respect to temperature - if you start off with a temperature of minus 4 and the temperature goes down by a further six; degrees then the temperature goes down to minus ten.

Likewise

- 4 - 6 = - 10



example

A slightly different problem would be

4 - 15 = ?

Again, think of what happens with respect to temperature - if you start off with a temperature of plus 4 (i.e. above freezing) and the temperature goes down by fifteen degrees then the temperature goes down to minus eleven.

Likewise

4 - 15 = - 11


Quick Quiz     Calculate a)   9 - 6   b)   6 - 9   c)   3 - 11   d)   5 - 7   e)   7 - 4   f)   - 9 - 5   g)   - 2 - 3  


By Ciara Leeming
3/11/2007

A LOTTERY scratchcard has been withdrawn from sale by Camelot - because players couldn't understand it.

The Cool Cash game - launched on Monday - was taken out of shops yesterday after some players failed to grasp whether or not they had won.

To qualify for a prize, users had to scratch away a window to reveal a temperature lower than the figure displayed on each card. As the game had a winter theme, the temperature was usually below freezing.

But the concept of comparing negative numbers proved too difficult for some Camelot received dozens of complaints on the first day from players who could not understand how, for example, -5 is higher than -6.

Tina Farrell, from Levenshulme, called Camelot after failing to win with several cards.

The 23-year-old, who said she had left school without a maths GCSE, said: "On one of my cards it said I had to find temperatures lower than -8. The numbers I uncovered were -6 and -7 so I thought I had won, and so did the woman in the shop. But when she scanned the card the machine said I hadn't.

"I phoned Camelot and they fobbed me off with some story that -6 is higher - not lower - than -8 but I'm not having it.

"I think Camelot are giving people the wrong impression - the card doesn't say to look for a colder or warmer temperature, it says to look for a higher or lower number. Six is a lower number than 8. Imagine how many people have been misled."

A Camelot spokeswoman said the game was withdrawn after reports that some players had not understood the concept.

She said: "The instructions for playing the Cool Cash scratchcard are clear - and are printed on each individual card and in the game procedures available at each retailer. However, because of the potential for player confusion we have decided to withdraw the game."

More than 15m adults in Britain have poor numeracy - the equivalent of a G or below at GCSE maths

Almost three times as many UK adults (15.1m) have poor numeracy - the equivalent of a G or below at GCSE maths - than with poor literacy skills, according to the government's Skills for Life survey.

Peter Hall, of the Association of Teachers of Mathematics, said: "The concept of minus numbers is something we would cover with 11 or 12 year olds, and we would expect them to have come across it before.

"The concept of smaller numbers is something that some people do seem to struggle with. Seven is clearly smaller than eight, so they focus on that and don't really see the minus sign. There is also a subtle difference in language between smaller - or lower - and colder. The number zero feels lower.

"There have always been some people who find numbers and basic mathematics difficult. Maybe in the past it was less noticeable because people could find jobs they could excel in without having qualifications in maths."

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Clash of Binary Operators in Addition/Subtraction

By the term 'binary operators', I refer to the operators

+   -

When using negative numbers, a clash of binary operators can occur, e.g.

\[ 2 + - 3 = ? \]

\[ 6 - - 5 = ? \]

More commonly, brackets would be used in the above e×pressions to increase readability, e.g.

\[ 2 + ( - 3) = ? \]

\[ 6 - ( - 5) = ? \]

When there is such a clash of binary operators, the following rules apply to reduce two "clashing" binary operators to just one binary operator

\[ - \ + \ = - \]
\[ + \ - \ = - \]
\[ - \ - \ = + \]

that is to say : a $-$ will dominate over a $+$ always, but when two $-$ clash, they becomes a single $+$

For example, an e×pression like

2 - ( - 6 ) = ?

would, by reference to the above table, simplify to

2 + 6 = ?

Likewise an e×pression like

5 + ( - 3 ) = ?

would become

5 - 3 = ?

A physical analogy for an expression like

- 3 + ( - 5 )

could be a bank account, where originally you had an overdraft of 3 euros (-3 euros) and you added another withdrawal of 5 euros. This is obviously equivalent to subtracting the 5 euros from the original amount, so could just be represented by

- 3 - 5 = - 8

Quick Quiz     Calculate :   
a)   7 + - 6  
b)   8 - + 3  
c)   24 - - 8  
d)   5 - (+7)  
e)   7 + (-4)  
f)   - 9 + - 5  
g)   - 2 - + 3  
h)   - 4 - (-7)  

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Multiplication and Division


Multiplication and Division involving negative numbers follows analogous rules as for addition/subtraction (in the previous section). These rules are summarized as follows

( - )( + ) = ( - )
\[ \frac{( - )}{( + )} = ( - ) \]

( + )( - ) = ( - )

\[ \frac{( + )}{( - )} = ( - ) \]

( - )( - ) = ( + )

\[ \frac{( - )}{( - )} = ( + ) \]


examples

2 × ( - 6 ) = - 12

( - 2 ) × 6 = - 12

( - 6 ) × ( - 2 ) = 12

\[ \frac{- 6}{3} = - 2 \]

\[ \frac{6}{- 3} = - 2 \]

\[ \frac{- 6}{- 3} = 2 \]

Quick Quiz     Calculate :   

a)   7 × (- 6)  
b)   8 × (+ 3)  
c)   - 4 × - 8  
d)   - 5 × 7  


e)   \[ \frac{8}{(-4)} \]  
f)   \[ \frac{- 25}{- 5} \]  
g)   \[ \frac{- 21}{3} \]  
h)   \[ \frac{- 42}{-7} \]  

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Addition / Subtraction calculations involving more than two numbers

Consider a calculation like

6 - 3 - 5 = ?

This can be carried out from left to right. Since

6 - 3 = 3

The calculation reduces to

3 - 5 = ?

which has the answer

-2

Alternatively

Given

6 - 3 - 5 = ?

Since

- 3 - 5 = - 8

The calculation reduces to

6 - 8 = ?

which has the answer

-2

as before

Quick Quiz     Calculate :    a)   7 - 6 + 3   b)   8 - 9 + 4   c)   - 24 - 8 - 3   d)   5 - 7 + 8  

e)   - 7 - 4 - 5   f)   - 9 + 6 + 12   g)   2 - 3 + 6   h)   - 4 -7 + 24  

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Mixture of Addition / Subtraction operations and Multiplication/Division operations

When you have a mixture of binary operators, the following rules apply

Multiplication / Division operations are carried out before Addition / Subtraction operations.

So often there are definite rules about the order in which you can tackle a problem. In practise, brackets will often be used to make the order of calculation clear (see below).

So in a calculation like

2 + 5 × 6 = ?

The first step is to do the multiplication first to produce

2 + 30 = ?

giving the answer

32

Quick Quiz     Calculate :    a)   7 × 6 - 5   b)   18 - 15 / 5   c)   18 - 3 × 5   d)   - 5 - 7 × 3  

e)   8 / 4 + 3   f)   - 9 × 5 - 7   g)   - 2 - 3 × 4   h)   4 × (-7) + 21  

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Brackets

Brackets are an important means of showing the order in which a calculation can be carried out. In a nutshell -

all calculations within brackets are done first.

Examples

\[ \frac{24}{( 5 + 3 )} = \frac{24}{( 8 )} = 3 \]

( 6 + 2 ) × 3 = ( 8 ) × 3 = 24

\[ \left( \frac{25}{5} \right) + 3 = (5) + 3 = 8 \]

( 3 - 4 ) × ( - 3 + 5 ) = ( - 1 ) × ( - 2 ) = 2

\[ \frac{( - 3 - 9 )}{( 2 - 5 )} = \frac{( - 12 )}{( - 3 )} = 4 \]

( 3 × 6 ) - ( 5 - ( - 4 ) ) = ( 3 × 6 ) - ( 5 + 4 ) = ( 18 ) - ( 9 ) = 9

Brackets can also be 'tiered', in which case it can help clarity if brackets of different shapes were used for each tier. For example

[ ( -9 ) × ( -5 ) × 2( 4 - 6 ) ] × [ 3 × 5( 7 - 4 ) × ( - 2 ) ]

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More on negative numbers, from the BBC site




Questions from Past Papers



Maria is doing research into food storage temperatures for her catering course. She finds the following table in a book.

Her domestic refrigerator has a fault and is $4^\circ C$ higher than the ideal temperature. What is the difference in temperature between her domestic refrigerator and the three star rated deep freeze if the deep freeze is at the correct temperature?

  • A    $17^\circ C$
  • B    $19^\circ C$
  • C    $25^\circ C$
  • D    $27^\circ C$

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