When people are asked what mathematicians do, the usual answer is that "they work with numbers". Although mathematicians nowadays feel slightly insulted by this answer, and many would deny they ever touch something so mundane as numbers, there is still a certain amount of truth in this answer. Historically mathematics begins with numbers and shapes---geometry, and investigating numbers and shapes is what mathematicians did over centuries.
To be more presise, early cultures usually don't know decimal numbers, not even fractions, but start with natural numbers. These are the numbers 1, 2, 3, 4, ... . Of course, when you start counting you need them. Some say the number 0 would also belong to the natural numbers, but historically the "0" comes much later, only about 600BC arabian mathematicians introduced it. That's also reasonable: If you have no cow, then you don't need a number to express that sad fact. And the negative numbers are also not natural.
For practical purposes, addition and multiplication were soon invented. If you have 34 sheep on the North meadow and 51 on the South meadow, how many do you have together? Or if each of your ship requires 17 sailors, and you have 7 ships, how many sailors do you need. The same with subtraction, although there were suddenly questions without an answer, like the famous: "You see three men enter a house. After that, five leave the house. How many are in the house?" If you have 36 cows and give your neighbor 13 cows, you have 24 of them, but what if you would give your neighbor 50 cows? The same with division. You can divided your 36 cows evenly on your three children, but not your 85 sheep. Of course, this is where fractions are required, and they emerged here as well, but still mathematicians began to investigate why certain numbers can be divided by some numbers. Maybe mathematics began to loose the total connection with practical purposes at that point, and something as impractical as prime numbers resulted. Or can prime numbers be applied somewhere?
This concept, as well as the concept of prime numbers, only applies to natural numbers. We say that a natural number A divides a natural number B
Since every divisor of a number is less or equal than the number, every number n can only have at most n divisors. For instance, the divisors of 15 are 1, 3, 5, and 15. The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Every number n (except 1, where both coincide) has at least two divisors---1 and n. Numbers with only two divisors are called prime numbers and investigated later.
Different from divisors, every natural number has infinitely many multiples. These multiples of n are exactly n, 2·n, 3·n, 4·n, ... .
A number is a common divisor of two other numbers if it is a divisor of both. In the same way, a number is a common multiple of two numbers if it is a multiple of both. For instance, 6 is a common divisor of 36 and 60. 12 a another common divisor of 36 and 60. The numbers 10, 100, 1000, 10000, ... are all common multiples of 2 and 5.
Since every number has only finitely many divisors, two given numbers a and b also have only finitely many common divisors. Therefore there is a smallest and a largest among them. The largest one is especially interesting and is called the greatest common divisor of a and b and written as GCD(a,b). For instance, GCD(36,60)=12, since the common divisors of 36 and 60 are the numbers 1, 2, 3, 6, 12.Multiples behave a little different. We have infinitely many common multiples, thus there is no largest one of them. But since all these common multiples are natural numbers, there still must be a smallest one of them, a least common multiple, denoted by LCM. Take LCM(6,9)=18 as example.
Theorem: For every two natural numbers n and m, GCD(n,m)·LCM(n,m)=n·m.
You may have observed that the GCD of two numbers can be found if you can factor both numbers into prime numbers, but this factorization itself is less than easy for large numbers. Still we may want to be able to find the GCD of two arbitrary numbers fast. The question is how? This problem was resolved by the famous Greek mathematician Euclid. He gave a procedure, a method, how to find the GCD of any two numbers in reasonable time.
Euclid's Algorithm (procedure)
for computing the GCD.
The input are two natural numbers a < b.
We divide b by a and get a remainder r.
If this remainder is not equal to 0, we rename
a as b1, rename r as a1,
and divide b1 by a1
to get another remainder r1.
We continue until eventually we must
(since r > r1 > r2 ...)
obtain a remainder
rm=0 (when dividing bm by am).
The last nonempty remainder rm-1 is the GCD of a and b.
There is also a version of this algorithm to find common units of two straight lines. We check how often the shorter one fits into the larger one. Then we proceed with the remaining part (remainder) and the former shorter line, and so on. Different from above, the process doesn't have to terminate and could in principle go on forever (if there is no common unit length contained in both lengths).
By the way, how would we find the LCM of two numbers quickly?
Is the square of an even number always even? Note that is not sufficient to present a list of examples like 22=4, 62=36, 182=364, 1022=10404. It could be true in most cases, with a few unknown exceptions.
Why would we even care? If somebody faces an even number and wants to know whether ist square is even or not, why should he or she not just square it and look at the result? Well, with the theorem known to be true, the square does not have to be computed, we would know that the square is even even before. More important, theorems are building blocks of other theorems. We will later see that we need this fact, that the square of all even numbers are even and the squares of all odd numbers odd, in the proof of the Theorem that the square root of 2 is not rational. In the proof, we don't look at concrete numbers but rather on variables which could be any numbers. Therefore we need the validity of our theorem also for any, for arbitrary numbers.
Proof:Let n be an even natural number. Remember that this means that there is another natural number m such that 2·m=n. Then n2=(2·m)2=4·m2= 2·(2·m). Since 2·m is again a natural number, this means that 2 is a divisor of n2, therefore n2 is again even.
Note how the proof didn't deal with concrete numbers. Rather the variable n was introduced, with arbitrary but even value. A statement like "The square of 14256 is even", although true, is not considered to be a general mathematical theorem. It lacks the generality. Mathematical theorems usually make statements about infinitely many numbers or mathematical objects.
Next we wonder about squares of odd numbers. First we would get data, meaning square some small odd numbers to get some idea. Obviously, since 12=1, 32=9, 52=25, 72=49, 112=121, it seems that the square of an odd number must also be odd. Isn't it true that if n2=n· is even, then n also has to be even? Or isn't it even true that a product can only be even if one of the factors is even? Yes, it is, and maybe we proof this fact first, since we would need it as building block of the odd squares theorem:
Proof:... ...
Class Activity: We can rephrase Theorem 2 as follows: If n divides the product n·m, then 2 divides n or m (or both). Can this be generalized? Is it true for all natural numbers k, n, and m that if k divides the product n·m, then it must also divide either n or m (or both)?
Proof:Using Theorem 2, the proof is rather easy.
The method of proof is called
"Reductuo Ad Absurdum",
you assume the opposite of what you want
to prove correct and show that this assumption logically
leads to something obviously wrong.
Let n be an odd number, and assume its square n2=n·n is not odd,
meaning, it is even. By Theorem 2, then n or n is even. But n cannot be even and odd
at the same time, we have a contradiction.
That means that the opposite of the assumption
must be true, therefore n2 is odd.
Here is another theorem with proof:
How are Theorems obtained? This is the most frequent way: