2.1 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 is addition, subtraction, multiplication and division (and perhaps also uglification and derision according to Lewis Carroll who was a well known mathematician).

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.

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.

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 , 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 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.