Algebra/What is math, exactly?

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Of all the subjects taught in schools throughout the world, Mathematics is possibly the one which has collected the image of being most feared and disliked.

So first off, what is math? What makes it so different from other areas of study, like languages or history? And more importantly, why on earth do you need to know anything about it?

[edit] Definition

Simply put, mathematics is the study of patterns -- finding patterns and explaining why such patterns exist. Patterns can be everywhere: shapes (what is the area of a 3-sided shape, 4-sided, 5-sided, any number of sides, what is the volume of solids? What solids are there? How many ways are there to rotate a solid so that it looks the same?), counting (How many ways are there to arrange 2 objects, 3, 4, any number? How many ways are there to arrange them in a circle? How many edges are there on a line? a square? a cube? a four-dimensional cube? How many ways are there to connect 5 dots? more? It is possible to connect 4 dots without any of the lines crossing. But what if we live on the surface of a sphere? Then how many can you connect together with none crossing? How many sudokus are there?) and so many more.

One particularly interesting class of patterns is the patterns of numbers. Whole numbers, for example, are deceptively simple looking: 1,2,3... everyone knows what they are, everyone is familiar with their addition and multiplication. But there are subtle and profound patterns lurking. For example, we can look at what numbers are formed when we repeatedly add 2. 2, 4, 6, 8.... We can look at what numbers are formed when we repeatedly add 3: 3, 6, 9, 12..... It's easy to see that every number falls into at least one of these sequences. But how many does it fall into, and which ones? For example, 12 is in the sequence 2, 4, 6, 8, 10, 12..., 3, 6, 9, 12, ..., 4, 8, 12..., 6, 12, ..., and 12..... What numbers fall into only one sequence? That's an extremely delicate and tricky question. The first few are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. How many of them are there? 3 and 5 are 2 apart, so are 5 and 7, and 11 and 13, and 17 and 19. How many pairs are there like this? These particular numbers are called "prime numbers."

There are all sorts of questions to ask about these simple sequences, and some of them lead to very profound statements about the structure of numbers.

But why are there such patterns? Why is it that there have to be an infinite number of primes? It's possible to imagine that this stream of numbers just stops, maybe after billions and billions of primes, maybe there are so many that people will never be able to compute them all -- but why is that impossible? Why is it that if you imagine that, then, if you are very imaginative, you will see that it is absurd?

The asking of questions about pure patterns is mathematics, as well as answering why it must be that way.

Now, why would you want to know about it? Simple. Because it is fun and because it is interesting.