Math, The Way It Should Be (Unit 1.4: Numbers, Part 4)

If you haven’t seen this series before, you should start here or here.  The last lesson covered rational numbers and how to solve equations involving multiplication and addition.  It also introduced another type of problem, those involving exponents.  Exponents, generally speaking, indicate multiple instances of multiplication.  In other words,

5² = 5·5 = 25

5³ = 5·5·5 = 125

It continues on like this for all integers and rational numbers.  The problem is when you want to go in the other direction.  To do this, you have to use an operation called a root.  The square root of x looks like this:

For equations like the one below, it is easy to find the solution of using a square root.  You just have to keep one thing in mind: a square root generates two answers, a negative and a positive answer.  This is because you are doing the opposite of multiplying two numbers together and the product of two negative numbers is positive.

The answer in this case is either 5 or -5!  If you check this result, you will find that squaring either number results in 25.  There are two problems with the square root, and they come when you encounter two different numbers.

In the first case, you try to take the square root of a number like two.  You can’t choose a rational number that, when multiplied times itself, results in two.  So we need a new kind of number.  These numbers are called irrational.  They cannot be written as the ratio of two integers and none of the previously mentioned numbers are irrational.  The irrational numbers are an ordered set and dense, but that’s the end of their nice properties.  When represented as decimals, they have an infinite number of digits after the decimal point and never repeat.  I try to avoid representing irrational numbers as decimals, instead preferring to solve equations in this way:

Now we can solve equations involving exponents with all positive numbers.  But what happens if we want to take the square root of a negative number?  I already said that the product of two positive or negative numbers is always positive.  So how do you choose a number that, when multiplied by itself, is equal to a negative number?  We introduce a new set of numbers, the imaginary numbers.  I’m not kidding.  That’s what they’re actually called and the most important imaginary number is i.  i is the square root of -1.  This allows us to factor negative numbers into -1 and a positive number.  We then take the square root of the positive number and multiply by i:

Don’t forget that there are still two answers!  When combined with the real numbers (integers, rational and irrational numbers), the imaginary numbers form the complex plane.  A complex number is one that is the sum or difference of a real number and an imaginary number.  Much farther down the line, we’ll see why it’s called the complex plane, but suffice it to say that it’s called that as a result of the complex plane not being an ordered set.  Now we’ve talked about all of the relevant numbers.  We can solve equations involving addition, subtraction, multiplication, division, and exponents for all real and imaginary numbers!

Homework:  as long as you’ve followed this lesson, I’m happy.  For the advanced student, the following exercises can be enlightening:

• Prove the square root of two is irrational (hint, recall the definition of rational and prove by contradiction)
• Prove that the complex plane isn't an ordered set (hint, recall the properties of an ordered set and prove by contradiction)

-Lane