Positivity

So far, the operations that I've discussed treat positive and negative numbers in essentially the same way. Certainly, there are rules for these operations —even quite useful ones such as ‹negative times negative is positive›— that mention positive and negative numbers, but we actually understand these rules by honestly trying to treat positive and negative numbers in the same fashion as much as we can. In contrast, sometimes we really need to treat positive and negative numbers separately. That is, some operations on real numbers are positivity operations, which act on the sign of a real number: whether that number is positive, negative, or zero.

Absolute value

The most fundamental positivity operation is taking the absolute value. Recall that the opposite of a real number is given by the point on the number line which is the same distance from 0 but in the opposite direction. Similarly, the absolute value of a real number is given by the point on the number line which is the same distance from 0 but in the positive direction. 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 is positive (or zero), |x| = x.
x is negative (or zero), |x| = −x.
But in general, you only know that |x| is either x or −x, and you can't say which. You can, however, say that |x| is either positive or zero; whatever x itself may be, its absolute value |x| is never negative.

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, 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.

Image: Real number line from −10 to 10.

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.

You can also compare complicated expressions built using the various operations. For example, to compare 3 + 4 and 2 · 5, you calculate that 3 + 4 is 7, while 2 · 5 is 10; since 7 < 10, you can conclude that 3 + 4 < 2 · 5. In summary:

For a more complicated example, compare |3 − 4| with 2 · 3 − 5:

You will do such comparisons often throughout this course.
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