Think of a number, any number. When someone says that to us, we tend to think of a natural number like 7. After all, these numbers are old friends. We know them since childhood. They are also the basis for all mathematics. So let’s begin our discussion of numbers with them, with the integers. When you take these positive integers, being sure to include the number 0, you get a collection that is called the set of natural numbers. I’ve written it here. The capital N is in what is called the blackboard style. You can see that I’ve put the numbers inside curly brackets, or braces as they’re called. That’s so they don’t escape. I’ve also used the word set.
We’re going to be talking about sets a lot, so let’s review basic set notation to make sure we’re all on the same wavelength. Suppose that A and B are two sets of numbers, say. What do we mean by containment? A is contained in B means that every element that’s in A is also in B. Intersection. A intersect B is the set of all elements common to both A and B. Now this raises a question. What if there are no numbers common to A and B? In that case, A intersect B is the empty set. Union.
The union of A and B is the set of all the elements in either set, and set difference– A minus B or A delete B, as it’s also read– is the set whose elements consists of all numbers in A but having removed those that are in B. For example, suppose E is the set of even integers in N. Then, the set difference N delete E is the set of odd numbers in N. Integers play an important role in a proof technique called mathematical induction. Here’s the way mathematical induction works. You want to prove a certain proposition or assertion whose formulation depends upon a positive integer n. And suppose we know the following.
For a certain n bar, the proposition P n bar is true. And suppose we also know that whenever P n is true, then the next proposition, P n plus 1 is true. It then follows that P n is true for all n beyond n bar. That’s a proof technique. It’s true that proofs will not be the main thing that we’re going to focus on in this course, but some experience with logical reasoning is very useful later on, for example, in calculus. Here’s an example of the use of induction. It’s a formula used by the great applied mathematician Archimedes well over 2,000 years ago.
It asserts that the sum of the first n squares is given by a certain formula in terms of n. That’s the proposition P n. Let’s check it for n equals 3, for example. In that case, the formula asserts that the sum of the first three squares is equal to a certain expression which I’ve obtained by replacing n by 3 in the general formula. You work out what it gives you, and you get 14. So that shows the formula is true for n equals 3 because the left side is the sum of 1 plus 4 plus 9, which is indeed 14. Now, we’re going to prove the general formula later on. But first, let’s discuss divisibility.
If you have two positive integers p and q, we say that p is divisible by q if it’s possible to write p as the product of q with another integer. Here are some facts about divisibility. An integer is divisible by 2– that is, is an even number– if and only if its final digit is divisible by 2. Do you notice the funny word there, i-f-f? That’s a standard mathematical shorthand for the phrase if and only if. It gives you the equivalence, complete equivalence of two assertions.
To return then to divisibility, an integer is divisible by 3 if and only if the sum of its digits is divisible by 3, and by 5 if and only if its last digit is a 0 or 5. Let’s apply these rules to the number of 714. It’s divisible by 2 because 4 is divisible by 2. It’s divisible by 3 because the sum of its digits is 12. And it’s not divisible by 5 because it doesn’t end in a 0 or a 5. Now, another concept we need is that of prime numbers. Incidentally, large prime numbers play a crucial role in the encryption codes these days that give you digital security. Obviously, any integer is divisible by itself and by 1.
Well, if you have an integer greater than 1 that is only divisible by itself and by 1, we say it is prime. For example, we can check that 2, 3, and 5 are prime numbers, but 15 isn’t because 15 is divisible by 3, for example. Some facts about prime numbers. There are infinitely many. Not that easy to prove, but if you find the right clever argument, not necessarily that hard either. Another fact. Every integer can be factored in a unique way into a product of prime numbers. That’s called the prime factorisation theorem. Let’s see how it works for the number 132. Obviously, that’s not a prime number. It’s divisible by 2.
I factor out a 2, and I get a 66. In turn, the 66 produces another factor of 2. Is the 33 a prime number? No. It’s equal to 3 times 11. And now every factor I have there is prime. So this is the prime factorisation of the number 132.