Absolute value

The absolute value of a real number is an important tool, which we will define now. Let x be any real number. Its absolute value is defined as follows. It’s equal to x if x happens to be greater or equal to 0. Otherwise, it’s equal to minus x when x is negative. It’s easy to deduce from this very definition that the absolute value of x is always positive. Furthermore, that it’s equal to 0 if and only if x is equal to 0. For example, then, we have the absolute value of minus 4 is the same as the absolute value of 4, namely 4. Here are some rules for the absolute value.

The absolute value of a product is the product of the absolute values. It’s easy to prove that right from the definition. A similar rule holds for quotients. Absolute x over y is absolute x over absolute y. Of course, as usual, in such a rule we must exclude the case y equals 0. As for the sum, we don’t have any equality. We have a famous inequality. It bears a name. It’s called the triangle inequality. It’s easy to see that this inequality is actually strict whenever x and y are of opposite sign because there’s cancellation on the left that doesn’t occur on the right.

For example, if you take y equals minus x, then you have 0 on the left, and you generally have something strictly positive on the right. Note. A common error that beginners make is to suppose that, if x is less than or equal y, then absolute x is less than or equal absolute y. Absolutely untrue. I urge you to find an example. Absolute values can often be interpreted in terms of distance, and I urge you to grasp the geometry of this because it’s very useful to make certain things obvious. First observation. Here’s the real line. Here’s a number x. What is this distance? Remember, we agreed that it was x.

That’s the same as absolute x because x is positive in this example. What if x is to the left of 0, a negative number? In that case, the distance is still a positive number, but it’s minus x since x is a negative. In other words, the distance is the absolute value of x. To summarise then, in all cases absolute x can be interpreted as the distance from 0 to x.

We can deduce from this certain things rather easily. Consider for example the inequality absolute x less than or equal to r. This says the distance from x to 0 is no bigger than r. Clearly, that means, by distance, that x has to be between minus r and r. That is, it’s the interval described here in red, or in our previous notation the closed interval minus r, r. Another interpretation of the absolute value is for the distance between any two points on the real line, a and b, say. The distance between them is the absolute value of b minus a, or equivalently, a minus b. It’s the same thing.

Now, pi is a famous number, and it doesn’t belong to the set of rationals. I say another famous number because we also saw the square root of 2 that didn’t belong to the rationals. We can talk about pi by ordering another pizza, a round one this time. Let’s slice it along the diameter, as it’s called, and this allows us to give the definition. Pi is the circumference of the circle– that’s the length of the circle– divided by the diameter of the circle– that’s the length of the black segment you see there. In calculus, you learn how to calculate pi to any precision you like, and here is a decimal representation of the number pi.

You don’t usually calculate it to that many decimals, but what the heck. Now, any real number has a decimal representation, and sometimes they’re very easy to find. For example, the number 5/2, a fraction, is equal to 2 and 1/2, which is equal to 2 plus 5/10, which by definition is 2.5. You can put more zeros after it, an infinite number, but you don’t have to. The number 1/3, which is a rational number, has an infinitely repeating decimal expansion. The 3 goes on forever. Now, with pi, there will be no repeats because it’s not rational, not even in blocks. It’s sort of random, the decimals that you get when you go further and further.

They have been computed to billions of places. Let’s not do that, but let’s instead try and study the significance of what these decimals mean. For example, pi begins with 3.14. What does that signify? Well, it means that pi is certainly greater or equal to 3.14 because there are other digits after that. But pi is no bigger than 3.15, which is 3.14 plus 1/100, 100 being 10 squared. In other words, in terms of absolute value, it tells that the distance between pi and 3.14 is less than 1/100. Similarly, if we take one more decimal place, 3.141, we will get the conclusion that pi is pretty close to 3.141. How close?

The distance will be less than 1/10 to the power 3, 1/1,000. And it goes on like that for every decimal place. We zero in on what the value of pi is closer and closer. If we go to six decimal places for example, the conclusion will be that the value of pi is the indicated number, and the error is less than 1/1,000,000.