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Basics of Algebra: Part IV

Author: oLahav

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Algebra, part IV- introductory number theory, complex numbers

This series of lessons is designed to help you learn, or review, the fundamentals of algebra. This lesson covers some number theory concepts, and introduces complex numbers.

Algebra opens the door to many other areas in mathematics. Number theory and complex analysis are great examples. Let's see what they're all about.

First of all- division and mods

division algorithm tells us that any number x can be divided by a number y such that x=yq+r

for some quotient q and remainder r. If r=0, we say x is divisible by y. In that case, y is called a factor of x. You can see why this makes sense with our earlier discussion of factoring.

We can also write this division statement as $x\equiv r \bmod { y}$ . This is saying the exact same thing as x=yq+r

. Note that unlike the equation, the mod form is not unique- you can have more than 1 value for x. For instance, if $x \equiv 3 \bmod 5$ , x can be 8, 13, 18, etc.

Now comes the fun part- primes!

A prime number is a number which has only 2 positive factor- 1 and itself. 2 is a prime number, as are 3,5,7,11,13,17,19,23 and so on and so forth. Every number that isn't prime is called a composite number.

According to a theory by Euclid (an old-school mathematician who did tons of cool work) and similar theories by other people (a respectful number of whom were Chinese), every number factors into a unique set of primes. For example, 12=2 * 2 * 3 , and no other number equals .

This is the basis for Number Theory, which obviously involves much more complex stuff. But I think in terms of algebra, knowing the basics is enough. And speaking of complex stuff…

Introducing: complex numbers, and "i"

We've already seen that most numbers you know are real, and rational. For example, x ^ 2=4

has two rational solutions, 2 and -2. However, we can't use rational solutions to solve x ^ 2=2

. The solution here is $\sqrt{2}$ , which cannot be expressed as a ratio of any kind. This gave rise to the concept of irrational numbers. You'll discover that the square-roots of all prime numbers are irrational.

But here's the kick - what do we do with stuff like x ^ 2= -1

? If you try the quadratic formula, you'll get a value that's negative under your root, and we can't take the root of negative numbers, so what do we do?

We turn to complex numbers. These are numbers that include the number $i=\sqrt{-1}$ . This is not a real number, since you can't draw it on any number line or anything like that, it's imaginary (yeah, real mathematicians actually use that term, not just Spongebob). Now we can solve every quadratic equation, even if the solution isn't real. Take x ^ 2+4=0

. The answer is

.

You should also note that if $i= \sqrt{- 1}$ , i ^ 2=- 1

, $i ^ 3=-\sqrt{ -1}$ and i ^ 4=1

.

p<>. And now you're ready to solve any quadratic equation that comes your way. With the algebraic tools you've acquired, you can now write equations, simplify expressions by factoring and expanding, and solve these equations, whether the solutions are real or imaginary.

And as always…

Thanks for reading this Welcome to Algebra Lesson!

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