Real numbers

When last we met, we had defined real numbers to be distances measured along the real axis, this infinite horizontal line where we’ve placed a 0, and then other numbers. For example, a number x on this line corresponds to the distance between 0 and x. Of course, numbers to the right are positive and have to be written with a plus sign. Those to the left are negative. Now, consider two real numbers, a and b. Terminology, we write a less than or equal b, if a is to the left of b on the real line, as these two numbers, for example, illustrate. We write a less than b, not less than or equal, when a is strictly to the left of b.

Notice that these two inequality symbols can be used in the other order as well, and they retain the meaning that they had, of course. Terminology, we say that x is positive if it’s greater or equal 0, so to the right of 0 on the line. And we say x is strictly positive if x is strictly greater than 0. The real numbers admit a complete arithmetic system that I’m briefly going to describe here. I’m going to describe it in symmetric terms with regards to addition and multiplication, which are the two main operations that we perform on real numbers. For addition, we have something called the associative law.

This means, as you see, that the result of adding three numbers, for example, is the same whether you add the first two first and then the third, or the second and the third then added to the first. And the same way for associative law. There’s also a commutative law. This says that the order doesn’t matter. So for example, the product ab is the same as the product ba. There is an identity element for both operations. For the case of addition, the identity element is 0, a plus 0 equals a, whereas for multiplication, it’s the number 1 that plays this role. There’s also a way to cancel out these operations. You cancel out an element by adding to it minus a.

You get 0. You cancel out in the sense that you come back to the identity element a by multiplying it by 1 over a, its reciprocal. Notice though that in this last law, you have to exclude the possibility a equals 0 because division by 0 is not defined. Another important arithmetic law is called the distributive law. It says that a times a sum can be broken down in the indicated way as a times each of the two terms added up. And finally, the vanishing of a product. If the product of two numbers equals 0, then one of them must be 0, or both, possibly.

So let us take two numbers, a and b on the real line, and ask you the following question. Where do you think the number a plus b over 2 sits on the real line? Answer? Right in the middle between a and b. A plus b over 2 is the average, also called the mean of the numbers a and b. It’s halfway between a and b. Now let’s look at intervals. Again, more terminology, more notation, but important.

Given, again, two numbers, a and b with a less than b, the interval ab– you notice I’ve used brackets here around a and b– is defined to be the set of all numbers– I also think of numbers as points since we’re on the line– between a and b including a and b themselves. Now, the set ab is called the closed interval. And you see that it’s given by the red segment that I’ve indicated on the diagram. If you don’t wish to include a and b in the interval, then we obtain the open interval. And it’s denoted by ab, but with the brackets turned the wrong way, so to speak.

Now, I have to point out that there is a big division in the world between those who write open intervals the way I’ve just indicated, and those who write open intervals using parentheses around the a and the b. As long as we know what we’re doing, either notation, of course, is fine. Now, there are also, of course, half open, half closed intervals, as I’ve indicated here. And they have their evident meaning. As for unbounded intervals, it requires us to introduce the symbol infinity.

The interval a infinity or plus infinity means the unbounded interval consisting of all real numbers to the right of a, all the way up to infinity, and including a itself since the bracket is turned the way it is. Similarly, minus infinity a is the set of all real numbers to the left of a, including a itself. Now, if a set of real numbers is called a, then the complement of a means the set of all those numbers that are not in a, notation a with a upper c on the right-hand side, c for complement, of course. Thus, the complement of a in another notation is the set difference r delete a.

To give a quick example, the complement of the interval a plus infinity is the interval minus infinity a. But notice that the a is not included in the complement. That’s why the bracket is turned the way it is. Terminology and facts now about the inequality relationship, basic arithmetic. A is less than or equal to itself. That’s called the reflexive law. A, the inequality relationship is anti-symmetric. That is, if a is less than or equal to b, and vice versa, then a and b must coincide. A third law is transitivity. If a is less than or equal b, and in turn, b is less than or equal c, then it follows that a is less than or equal to c.

What about the compatibility of inequality where there are two operations of addition and multiplication? Well, this first law says that if you have two inequalities in the same direction, then you can add them side by side, and you get a plus c is less than or equal b plus d. So you can add inequalities. The second observation is that you can add the identical number to each side of an inequality, and that preserves the inequality. And a third law here says that if you have a positive number c, then you can multiply the inequality across by c, and it remains true. It is preserved. Now I must warn you about something though.

It’s a mistake one commonly sees in the early days of this sort of course. What happens if you multiply across our inequality by a negative number? So, if c is a negative number, and a is less than or equal b, what does that tell me about ac and bc? Answer? It tells me the reverse inequality. That is, the inequality has actually been reversed. It’s easy to see on an example how this can happen and why it should happen. Here are two numbers, a and b. As you see, a is less than b because it’s on the left. If you now multiply by 2, where does 2b appear? Answer? Even more to the right. Where does 2a appear? Answer?

Even more to the left. And as you can see, the inequality is preserved. 2a is less than 2b. However, if you take negatives, what happens? Where is minus 2b? Well, it’s over here. And where is minus 2a? It’s over there. And now you see that the inequality is reversed.

Now, the study of real numbers will be continued in the next segment when we look at absolute value.