Arithmetic/Introduction to Radicals

What are radicals?
To find out what radicals are, let's look at what multiplication and division are. Multiplication is simply adding two numbers a certain number of times, i.e. $$5*4=5+5+5+5$$. Though we can't think of division in this way usually, we know that it is the "opposite" of multiplication.

This means that if we multiply some number by, say, 2, and then divide it by two, we will get that same number. It is in this way that multiplication and division "undo" each other. Mathematically, we could say $$x*a \div a=x$$, or "x times a, divided by a, equals x".

So, radicals are like division to exponents; in the same way that division undoes multiplication, and vice versa, radicals undo exponents. Radicals are usually written like $$\sqrt{x}$$.-

A focus on square roots
So, we know that $$3^2=9$$; this means, also, that $$\sqrt{9}=3$$. In the same way, $$\sqrt{16}=4$$, since 4 squared equals 16.

A focus on cube roots
In the same way, we know that $$2^3=2*2*2=8$$, thus $$\sqrt[3]{8}=2$$. Similarly, $$\sqrt[3]{27}=3$$.

Domains: What is the square root of 2?
You may be wondering how to take the square root of numbers like 2 or 3. Surely there isn't any integer that, when you multiply it by itself, you'll get two. So what could it be? Possibly a fraction?

Actually, no. The square root of two is the square root of two. There is no fraction that exactly equals it. The same is true for the square roots of 2, 5, 6, 8, etc.

Another look ahead at irrational numbers
There are actually many numbers that can't be represented as fractions (actually, there are as many of these numbers as those that can be represented with fractions). These numbers are called irrational numbers, since they can't be made into a "rational" fraction.

Domains: What is the square root of a negative number?
If you remember from the exponents chapter, we said that any number squared, to the fourth, or any even number, is positive. Square root finds that number, given its square. But what about numbers like -1, -2, or -543?

Well, we know that no rational, or irrational, number squared equals a negative number. That much is true. But, just like how we extended rational numbers to the irrationals as well when we tried to square root 3, we're going to have to extend our notion of rationals and irrationals (which, combined, make the "real" numbers). This group of numbers that, squared, gives a negative number is called the imaginary numbers.

Just as how $$\sqrt{2}$$ cannot be written as a rational fraction, $$\sqrt{-1}$$ cannot be written as a real number. It is simply $$\sqrt{-1}$$. For shortness, we call this number $$i$$, which stands for "imaginary" (since you can't have "i" potatoes, or "i" anything, really).