

When do we use negative numbers?  What
are directed numbers?  When do we use negative numbers? We use negative numbers in our everyday life without thinking about it. 'It was so cold last night, the temperature was 4 degrees below zero!' Another
way of saying '4 degrees below zero' is 4 ° C. 'Lake Eyre is 16 metres below sea level.' We refer to sea level as the zero point for measuring landforms, such as mountains, so on this scale the height of Lake Eyre is actually 16 metres. 'My bank account is in the red by $150!' 'In the red' is a common phrase used when we talk about banking, meaning that we owe money. If I am 'in the red' by $150 I actually have $150 in my account. (This means the bank has lent me $150 and eventually I will need to pay it back.) 'Jerry
made a loss on the horses of $20.' If Jerry made a loss of $20 on the horses, this means he has $20 less than before he started betting. What
are Directed Numbers? For every positive number there is an opposite negative number. For example, the opposite of 4 is 4 and the opposite of 239 is 239. This concept is illustrated well on a number line, see below in models for directed number arithmetic. Negative and positive numbers are useful to indicate direction either side of a zero reference point, the positive numbers indicate one direction and the negative numbers indicate the opposite direction. Because negative and positive numbers give direction they are often called directed numbers. While the term negative numbers is commonly used, in many cases the term directed numbers would be more appropriate. For example, if we are talking about 'negative number arithmetic' we often mean this to include arithmetic of both negative and positive numbers, i.e. directed numbers, therefore it is useful to be aware that the terms negative numbers and directed numbers are often used interchangeably. When we are only working with positive numbers we usually just refer to them as numbers. When we use directed numbers we are often referring to real contexts where opposites are involved. Can you think of others?
Models for directed number arithmetic There are a number of models which can be used to illustrate operations with negative and positive numbers. In this topic we will be mainly using integers, by this we mean whole numbers and the negatives of whole numbers. However we need to be aware that there are many other negative numbers other than the negatives of whole numbers, for example, 1.5, 1/2 and 3.45321. These negative numbers can be represented on the number line also, as can their opposites of 1.5, 1/2 and 3.45321, and follow exactly the same mathematical rules as negative integers. The only reason we are talking about integers here is because they are the simplest for a teaching context.
The
number line model
If we pick any number on the number line all the numbers to the left of it are smaller. We can also see that all numbers to the right of it are bigger. This model represents a number by a point on a line.
The Annihilation model A model which is useful for illustrating operations involving positive and negative numbers is the annihilation model. In this model a number is represented by a collection of chips. The annihilation model has positive chips and negative chips and works on the basis that a positive chip will annihilate a negative chip and vice versa. Sometimes the positive chip is black and sometimes the negative chip is red in analogy with colours used for credit and debit.
The table below shows the model in use. The positive and negative chips can be arranged to represent positive or negative integers.
Note that any number can be represented in many ways. Which
model is best?
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