# The real numbers

There are several different kinds of numbers, but in this course we work with the real numbers. Imagine a line, perfectly straight, infinitely thin and infinitely long. Also imagine that no matter how closely you look at this line, it's always a perfect line at every scale. (No line in the real world is like this, since any object in the real world is finite, made of atoms and molecules and the like. But this ideal line is often a good approximation for actual lines.) If you mark two distinct points on this line, calling them ‘0’ and ‘1’, then this line becomes the so-called real line.

The real numbers correspond to the points on the line. For example, the point marked ‘0’ is the number 0 (zero), and the point marked ‘1’ is the number 1 (one). Given these, you can find whatever other points you may wish, such as 2, 3, 17, −5, 7/9, 8.6, and so on. The direction from 0 to 1 is the positive direction, and the direction from 1 to 0 is the negative direction. The distance between 0 and 1 gives the scale on the line, also called the unit distance.

So, if you start at 0 and go in the positive direction for the unit distance, then you get to 1. If you continue from there in the same direction for the same distance, then you get to 2, then 3, 4, and so on. If you go in the negative direction instead, then you get to −1, then −2, −3, and so on. The numbers that you get in this way are the integers. But there are many numbers in between the integers; these are the fractional numbers.

The numbers in the positive direction from 0 are the positive numbers; and the numbers in the negative direction from 0 are the negative numbers. The natural numbers are simply the positive integers, and the whole numbers are the non-negative integers. So, the natural numbers are 1, 2, 3, …; the whole numbers are 0, 1, 2, 3, …; and the integers are …, −2, −1, 0, 1, 2, ….

Here is a number line with the integers from −10 to 10 marked. Notice the arrows on the end, indicating that the line should go on forever, even though obviously one can only draw a little part of it.

## Arithmetic operations

Traditionally, there are four basic arithmetic operations that we do with real numbers: addition, subtraction, multiplication, and division. Actually, it's probably more basic to talk about addition, multiplication, taking opposites, and taking reciprocals; then subtraction means adding the opposite, and division means multiplying by the reciprocal. There are several other operations as well, two of which —exponentiation and taking absolute values— will also come up in this course.

In terms of the real number line, addition involves moving along the number line. To add a number, first see how you must move along the line to go from 0 to that number. Then start at the number you're adding to, and make that same motion. For example, to go from 0 to 4, you move the unit distance 4 times in the positive direction. Thus, to find 2 + 4, you start at 2 and move the unit distance 4 times in the positive direction, ending at 6. From a geometric perspective, this is why 2 + 4 is 6.

Here is a picture of the calculation 2 + 4. (By the way, the artist forgot to put the arrows at the ends of this real line.)

The opposite of a real number is given by a point which is the same distance from 0 but in the opposite direction. For example, the opposite of 7 is −7, while the opposite of −4 is 4. Note that the minus sign (‘−’) in the names of negative numbers does not really mean negative; rather, the minus sign means opposite! Using this symbol, the examples above become as follows:

• −(7) = −7;
• −(−4) = 4.
In other words, a negative number has a minus sign in its name only because there is no simpler name without a minus sign; but the minus sign is not itself the reason that the number is negative. This is particularly important in algebra, because −x might be positive or negative, depending on what x is. For example, if x is −4, then −x is 4, as in the example above.

Now subtraction simply means adding the opposite. So for example, subtracting 7 is the same as adding −7, and subtracting −4 is the same as adding 4. To be more specific, suppose I want to subtract −4 from 2; that is, I want to calculate 2 − (−4). Then since the opposite of −4 is 4, this means the same as 2 + 4, which, as I mentioned above, is 6. In summary, a full calculation consists of these steps:

• 2 − (−4) — original problem;
• 2 + (−(−4)) — since subtraction means adding the opposite;
• 2 + 4 — since we found on the number line that −(−4) is 4;
• 6 — since we found on the number line that 2 + 4 is 6.
Of course, you don't really need to go through all of these steps; but this fully detailed calculation is the reason why 2 − (−4) is 6. (Notice that the minus sign still means opposite, even when it indicates subtraction. In that case, the minus sign just tells us to add the opposite.)

On the number line, addition has to do with how you move from 0 to a number, so it has a special relationship with 0. Multiplication, on the other hand, has a special relationship with the number 1. But rather than simply moving from 1 to the number you're multiplying by, you must expand and contract the number line instead. As you expand and contract the number line, the number 0 is fixed (so 0 is still special in multiplication, only in a different way). To be precise, to multiply by a number, expand and contract the number line, fixing 0, to move 1 to the number in question. Then whatever number you're multiplying by will move to your answer.

For example, to multiply by 3, expand the number line to 3 times its length, because this will move 1 to 3. Then the number 2 moves in the positive direction (getting even further from 0) to 6, so 3 · 2 is 6. On the other hand, the number −2 moves in the negative direction (also getting even further from 0) to −6, so 3 · (−2) is −6. As another example, to multiply by 1/2 (one half), you must contract the number line to half its length, because this will move 1 to 1/2. As you do this, the number 2 moves in the negative direction (closer to 0), becoming 1, while the number −2 moves in the positive direction (also closer to 0), becoming −1. Thus, (1/2) · 2 is 1, while (1/2) · (−2) is −1.

As a final example, to multiply by −3, you must contract the number line to 0, then expand it in the opposite direction to 3 times its length, because this will move 1 to −3. As you do this, the number 2 moves in the negative direction (past 0) to −6, while the number −2 moves in the positive direction (also past 0) to 6. Thus, (−3) · 2 is −6, while (−3) · (−2) is 6. Again, this is why multiplying a negative number by a negative number yields a positive result: every time you multiply by a negative number, you move every number past 0 to the opposite side, in particular moving negative numbers to positive numbers.

Next, the reciprocal of a number, if the number has a reciprocal, is whatever you must multiply the number by to get the result 1. So for example, since 2 · 1/2 = 1, the reciprocal of 2 is 1/2. (Actually, this is the reason why we call 1/2 ‘1/2’; see the next paragraph.) But notice that there is a number that has no reciprocal: there is no such thing as the reciprocal of 0! This is because, no matter how we expand and contract the number line in multiplication, the number 0 is always fixed, so there is no way that it will ever move to the number 1.

Finally, just as subtraction means adding the opposite, so division means multiplying by the reciprocal. For example, since the reciprocal of 2 is 1/2, 3 divided by 2 means 3 times 1/2. But we really have no better name for this than ‘3/2’, which literally just means 3 divided by 2. Nevertheless, if you want to find 3/2 on the number line, you can start at 1/2 (the reciprocal of 2) and then expand the number line to 3 times its length to find 3/2. As another example, to find 3/(1/2), that is 3 divided by 1/2, first notice that the reciprocal of 1/2 is 2 (since 1/2 · 2 = 1), so 3/(1/2) means 3 · 2, which is 6. Again, this is, in detail, why 3 divided by 1/2 is 6, even if you can do the calculation in another way.

It's important to notice that, because 0 has no reciprocal, it is impossible to divide by 0! We say that division by 0 is undefined, because nobody has given any meaning to an expression like ‘3/0’. Of course, the meaning of ‘3/0’ should be whatever you get when you multiply 3 by the reciprocal of 0, but there is no reciprocal of 0 in the first place. Notice that this problem does not arise with ‘0/3’; this means whatever you get when you multiply 0 by the reciprocal of 3, which is simply 0 (since there is a reciprocal of 3). So 0/3 is 0, but 3/0 does not exist. (Also, 0/0 does not exist either!)

## Types of numbers

I've already mentioned how some numbers are positive, while some are negative instead and 0 is neither; and I've mentioned how some are integers, how some of these (the ones that aren't negative) are whole numbers, and how some of these (the ones that are positive) are natural numbers. Here's another way to look at these, which continues to some other types of numbers.

The natural numbers are those real numbers that can be built out of:

• the number 1;
• multiplication (optional).
Multiplication is optional here; it's an important fact of mathematics that you can multiply by a natural number by simply adding a bunch of times; multiplication by natural numbers is repeated addition.

The whole numbers are those real numbers that can be built out of:

• the number 0;
• the number 1;
• multiplication (optional).
Multiplication is again optional; multiplying by 0 always results in 0.

The integers are those real numbers that can be built out of:

• the number 0;
• the number 1;
• taking opposites;
• multiplication (optional); and
• subtraction (optional).
Once more, multiplication is optional, because multiplying by a negative integer is repeated subtraction. That subtraction is optional here is trivial; subtraction is simply adding the opposite.

This idea can be extended one step further, bringing in reciprocals (and division). The rational numbers are those real numbers that can be built out of:

• the number 0;
• the number 1;
• taking opposites;
• multiplication;
• taking reciprocals (except the reciprocal of 0);
• subtraction (optional); and
• division (optional).
Now multiplication is not optional; you cannot multiply by a rational number using only addition and subtraction. But division is optional simply because it's multiplying by the reciprocal.

There are other types of real numbers; in particular, by the end of the course, I'll be able to describe for you the algebraic numbers. For now, I'll just say that an algebraic number is defined not by how you can build it up, but instead whether you can get back to 0 in a nontrivial way using that number, the number 1, and the basic operations of real numbers.

For most of these types of real numbers, there is a special term for a number which is not of that type. Here they are:

A number which is not is …
an integer,a fractional number.
a rational number,an irrational number.
an algebraic number,a transcendental number.

Nearly all of the numbers that we'll be using in this class are rational. There are many ways to write rational numbers, but the best way for the purposes of algebra is as common (possibly improper) fractions, like 1/2, 7/6, and −3/8. Less useful are mixed numbers like 1 1/6; this means 1 + 1/6, which is the same as 7/6. Also less useful are decimal fractions like 0.375; this means 375/1000, which is the same as 3/8. Finally, repeating decimals can also be turned into common fractions; for example, 1.1363636… means 1 + 1/10 + 36/990, which (if you work it out) is the same as 25/22. (Notice that an infinite decimal that does not repeat, such as 1.010010001…, is an irrational number.) Sometimes it's useful if your final answer is a mixed number or a decimal approximation, but within a calculation, common fractions are almost always the best.

## Order

Since real numbers are arranged in a line, we can consider which of two numbers come first in a given direction. If one number comes before another as we go in the positive direction, then this number is less than the other. And if one number comes before another as we go in the negative direction, then this number is greater than the other. Of course, the remaining possibility is that the two numbers come at precisely the same point; then these numbers are equal. Given any two numbers, exactly one of these situations can arise. For example, in the number line below (repeating a number line from earlier on this page), the positive direction is to the right and the negative direction is to the left (as is usual). Accordingly, since 2 is the left of 3, 2 is less than 3, which is written 2 < 3. Similarly, since 1 is to the right of −4, 1 is greater than −4, which is written 1 > −4. Finally, since 2 is of course the same as 2, 2 is equal to 2, written 2 = 2.

Given any two expressions for real numbers, exactly one of the three symbols ‘<’, ‘>’, and ‘=’ is appropriate to put between them. If you can see the numbers on a number line, it's obvious which symbol to use. Otherwise, you can calculate it by subtracting, as in this table:

If a − b is … then …
positive,a > b.
negative,a < b.
zero,a = b.
Here I've used a little algebra to summarise infinitely many different facts in a few lines. To use this, you would put specific numbers in place of the variables a and b. This is particularly useful in the case of fractions. For example, because 2/3 − 3/4 works out to −1/12, which is negative, you know that 2/3 < 3/4.

The symbols ‘<’, ‘>’, and ‘=’ give complete information about the relative order of two real numbers. Sometimes you only have partial information; then you can use the symbols ‘≤’, ‘≥’, and ‘≠’, as follows:

If …then … or …but not
a ≤ b,a < b, a = b,a > b.
a ≥ b,a > b, a = b,a < b.
a ≠ b, a < b,a > b, a = b.
For example, since 2 < 3, it is correct to say that 2 ≤ 3, and it's also correct to say that 2 ≠ 3, but it's wrong to say that 2 ≥ 3. Also, if you learn in an algebra problem that x ≥ 4, then it may be that x > 4 or that x = 4; you don't know which, but you do at least know that x < 4 is false.

## Absolute value

Besides the traditional four operations mentioned above (addition, subtraction, multiplication, and division) and the operations of taking opposites and taking reciprocals (which are used to define subtraction and division), there is another operation that we use in this course: taking the absolute value. Recall that the opposite of a real number is given by a point which is the same distance from 0 but on the opposite side. Similarly, the absolute value of a real number is given by a point which is the same distance from 0 but on the positive side. So the absolute value of a negative number is positive, and the absolute value of a positive number is also positive. (The absolute value of 0 is simply 0 again.)

The absolute value of a number is denoted by placing that number inside two vertical bars. So for example, |3| is the absolute value of 3, which is again 3. Similarly, |−3| is the absolute value of −3, which is also 3. Sometimes people think that the point of the absolute value bars is to remove minus signs, or to change minus signs to plus signs. But that is not true! For example, |2 − 6| means |−4|, which is 4, while |2 + 6| and 2 + 6 are both 8. This is especially important in algebra, where you cannot change |2 − x| to |2 + x| or 2 + x, nor can you change |2 + x| to 2 + x.

In particular, there is no way to change |x| to anything simpler, unless you happen to know whether x might be positive or negative. If you do know, then you can say this:

If …then …
x ≥ 0,|x| = x.
x ≤ 0, |x| = −x.
But in general, you only know that |x| is either x or −x, and you can't say which.
Go back to the course homepage.

This web page was written between 2007 and 2010 by Toby Bartels. Toby reserves no legal rights to it. The images on this web page are from Wikimedia projects. They are available under free licences; follow their links for more information about them.

The permanent URI of this web page is `http://tobybartels.name/MATH-0950/2010s/reals/`.