# 1.1 What Are Numbers? The Rational Numbers

We have lots of kinds of numbers but they all start with the **natural numbers**, which are
\(1, 2, 3\), and so on.

If you count your figures and toes, you will come to \(20\) (most of you will), and that is a natural number. We can, in our imagination, consider that these natural numbers go on forever, past a million, a billion, a trillion, and so on.

In elementary school you studied not only these numbers, but how you can perform operations on them.

**What operations?**

There are **addition, subtraction, multiplication** and **division**.

You can **add** two natural numbers together, and you will always get another natural number, as in
the famous fact that one and one are two.

Subtraction, on the other hand, is trickier. If you subtract a number, for example the number \(5\), from
itself, you get something new, something that is not a natural number at all. We call it the number \(0\) or
**zero**. And if you subtract a number, again say \(5\), from a smaller number, say \(3\), then
you get something else that is new, namely a negative integer, which in this case is \(-2\), called
**"minus two"**.

You can use numbers to count the number of pennies you have in your pocket. Thus you might have five pennies in your pocket. Zero is the number of pennies you would have if your pocket had a hole in it, and all those you put in immediately fell out again.

Now suppose you go to a store, and the storekeeper is foolish enough to give you credit. Suppose further that you had five pennies, and you bought some expensive item costing 11 pennies. Then the negative integer, \(-6\), represents the fact that not only do you have no pennies but if you got six more, you would be obligated to surrender them to pay for this item. Six here is the number of pennies you would owe your creditor, if you were to pay him your \(5\) pennies and he gave you the object, and lent you the rest of the money.

So to accommodate subtraction, and to be able to represent "amount owed" by numbers, we extend the natural
numbers to include the numbers \(0\) and the negatives of the natural numbers. This entire set of numbers,
positive natural numbers, their negatives and 0 is called the set of **integers**, and is denoted
by the letter **\(Z\).**

We can take any two members of **\(Z\)** and add them or subtract them and in either case get
another member of **\(Z\).**

**I know all that, but I am very rusty on actual additions and subtractions. I get them wrong much of the
time I try to do them.**

Most people will make a mistake roughly once in any ten additions or subtractions of single digits that they perform. This means that if they add or subtract numbers having many digits, like \(1234123\) and \(5432121\) they stand an excellent chance of getting the wrong answer.

Fortunately that is of no significance today. You can easily check additions and subtractions on a calculator or on a spreadsheet, and see if you get the same answer several different times. Unfortunately I usually make an error in keying in the numbers to add or subtract, or add instead of subtract or do something else equally absurd. All that means today is that I must do every calculation at least three times, to have a reasonable chance of correctness. True the amount of my effort is triple what it might be, but three times very little effort is still very little effort.

If you have this problem you will be best off adding or subtracting on a spreadsheet. Then you can look at your computation and use your judgment as to whether it makes sense. Here are some rules for checking for sense.

When you add positive numbers the result should be bigger than both of the two **"summands"** that
you added. If one of the numbers is positive and one is negative, the magnitude (the value if you ignore any
minus sign) of the sum should be smaller than the magnitude of the larger of the two, and the sign should be
that of the summand with the larger magnitude.

Also, the least significant digits of your numbers should add or subtract correctly, if you ignore the rest. For example, if you subtract \(431\) from \(512\) then the last digit of the answer had better be \(1\) which is \(2\) minus \(1\).

If your checking produces something suspicious, try your computation again, being more careful, particularly with the input data.

The operation of subtracting 5 from another number, **undoes** the operation of adding \(5\) to
another number. Thus, if you do both operations, add five and then subtract five, or vice versa, you are back
where you started from: \(3 + 5 - 5 = 3\).

Adding \(5\) and subtracting \(5\) are said to be **inverse** operations to one another, because of
this property: **Performing them one after the other is equivalent to doing nothing.**

**By the way, why isn’t \(0\) a natural number?**

I have no idea. That’s the way people defined natural numbers long ago, and nobody has cared much for changing that definition.

Back in elementary school you also encountered the notion of **multiplication.** This is something
you can do to two integers which will produce a third one called their **product.** You were (I
hope) forced to learn a multiplication table which gives the product of each pair of single digit numbers and
then learned how to use this table to multiply numbers with more digits.

**I was never very good at this** **.**

In olden days you had to be able to do these things, additions and multiplications, if only to be able to handle money and to perform ordinary purchases without being swindled.

Now you can use a calculator or computer spreadsheet to do these things, if you know how to enter integers and to push the \(+\) or \(-\) or \(*\) and = buttons as appropriate.

( *Unfortunately this fact has led pedagogues to believe they do not have to force pupils to go through the
drudgery of learning the multiplication table.*

*This does much harm to those who don't bother to do so, because of the way our brains function. It turns out
that the more time we spend on any activity as children, and even as adults, the bigger the area of the brain
gets that is devoted to that activity, and the bigger it gets, the better we get at that activity.*

*Thus, your spending less time learning the multiplication table has the effect of reducing the area of your
brain devoted to calculation, which impedes your further mathematical development.*

*Your skill at mathematics will be directly proportional to the amount of time you choose to devote to
thinking about it. And that is up to you.* )

Once we are acquainted with multiplication, a natural question is: how can we undo multiplication? What is the
inverse operation, say to multiplying by \(5\), so that multiplying and then doing it is the same as doing
nothing? This operation is called **division.** So you learned how to divide integers. The
**inverse operation to multiplying by \(x\) is dividing by \(x\)**, (unless \(x\) is \(0\)).

Now here comes a problem: if we try to divide \(5\) by \(3\) we do not get an integer. So, just as we had to
extend the natural numbers to integers to accommodate the operation of subtraction, **we have to extend
our numbers from integers to include also ratios of integers** , like \(\frac{5}{3}\), if we want to make
division well defined for every pair of non-zero integers. And we want to be able to define division wherever we
can.

Ratios of integers are called rational numbers, and you get one for any pairs of integers, so long as the second
integer, called the denominator, is not zero. Ratios like \(\frac{5}{3}\) which are not themselves integers are
called **fractions.**

Once we have introduced fractions, we want to provide rules for adding and subtracting them and for multiplying and dividing them. These start to get complicated, but fortunately for us, we have calculators and spreadsheets that can do these things without complaining at all if we have the wit to enter what we want done.

There is one thing we cannot do with our rational numbers, and that is to divide by \(0\). Division, after all, is the action of undoing multiplication. But multiplying any number by 0 gives result \(0\). There is no way to get back from this \(0\) product what you multiplied \(0\) by to get it.

Of course adding and multiplying (and subtracting and dividing) fractions is more complicated than doing so for integers. To multiply say \(\frac{a}{b}\) times \(\frac{c}{d}\), the new numerator is the product of the old ones (namely \(ac\)) and the new denominator is the product of the old ones (\(bd\)), so the product is \(\frac{ac}{bd}\): \(\frac{a}{b}*\frac{c}{d} = \frac{ac}{bd}\).

The inverse operation of multiplying by \(\frac{c}{d}\) is multiplying by \(\frac{d}{c}\), and that inverse is by definition the operation of dividing by \(\frac{c}{d}\). The product of any number and its inverse is always \(1\). This means that \(\frac{d}{d}\) is always \(1\) for any \(d\) other than \(0\).

Thus \(\frac{a}{b}\) divided by \(\frac{c}{d}\) is \(\frac{a}{b}\) multiplied by the inverse of \(\frac{c}{d}\) which is \(\frac{a}{b}\) multiplied by \(\frac{d}{c}\). The answer is \(\frac{ad}{bc}\).

Adding is a bit trickier. The notion of addition can be applied to objects as well as to numbers, in the following sense. We know, for example, that \(3+5\) is \(8\). That means that if we have 3 radishes and dig up \(5\) more, we will have \(8\) radishes (assuming nobody has eaten the first \(3\)). And the same is true for any other objects in place of radishes. This tells us how to add fractions that have the same denominator. Thus \(\frac{3}{a} + \frac{5}{a}\) is \(\frac{8}{a}\) in which \(\frac{1}{a}\) has replaced a radish. We are applying the general rule for addition of like things to the object \(\frac{1}{a}\).

To add fractions with different denominators you must first change them so that the denominators are the same, then add the numerators like you were adding numbers. The easiest way to do this is to make the new denominator the product of the old ones. Thus to find \(\frac{a}{b} + \frac{c}{d}\) you first multiply the first term by \(\frac{d}{d}\), and the second by \(\frac{b}{b}\), getting \(\frac{ad}{bd} + \frac{cb}{bd}\) and the answer is \(\frac{ad+cb}{bd}\). You can do the same sort of thing for subtraction.

You were probably forced to factor out common terms in the numerator and denominator in that answer in school, but you don’t have to do that in entering the answer in a spreadsheet, which makes addition of fractions much easier when you use spreadsheets.