Why is -9 not an irrational number?

Question

According to my algebra instructor, all irrational numbers are numbers that do not have square roots that are even? According to what I've found online, irrational numbers cannot be written as a simple fraction. -9 is a whole number and therefore can be converted into a simple fraction, correct? But it has no square root? CONFUSED!

Answer 1

-9 is not an irrational number due to a few things.

First, let's take a look at the definition of a rational and irrational number.

The definition of a rational number is as follows:

Any number that can be written as a fraction with integers

The definition of an irrational number is this:

A real number that cannot be expressed in the form of a/b, (where b cannot equal 0).  In decimal form, it either never terminates or repeats itself.

So let's take a look at -9 with these two definitions in mind,

What can we do to the -9 to make it a fraction?

We can divide it by 1 so now it looks like this, -9/1.

Which of the two definitions best describes this number now?

I hope that I have explained this adequately for your understanding.

Answer 2

The answer to your question on why -9 has no square root is further along in your studies, where you will learn to deal with the world of complex numbers.

The problem you are encountering is that there is no number that can be multiplied by itself to get -9, because if you remember multiplying like signs always give a positive number.

Not to leave you in the dark till then, the square root of -9, is 3i , the letter i letting us know it’s not a real number, in fact it's imaginary!

Most calculators are not set up to process imaginary numbers, so they simply put the word "Error" on the screen, which is why it seems to not have a square root.

So in all truth using -9 is not a great example for the proof of irrational/rational numbers, as it is neither.

(since the number to be squared to get -9, and the square root of -9 are not "real" numbers, and a number has to have a "Real" root to be part of the irrational/rational test you describe.)

 

Answer 3

∏I see where your confusion is coming from - there is a missing "piece" in your explanation from your instructor - a rational number, when squared, has even exponents. This is a difficult definition to follow, but lets see if I can give you an example. 

2 is rational; 2² is it's square, and it's exponent is an even number (2)

√2 is irrational; 2¹ is it's square, and the exponent here is an odd number (1)

√8 is irrational; 2³ (=8) is it's square, and the exponent here is an odd number (3)

The more common definition used is the one listed above by Katlin - a rational number is any number which can be written as a ratio, or fraction (a/b). 

Irrational numbers are made up of square roots (other than the square roots of perfect squares, like 4, 16, etc), higher order roots, the number pi, e, etc. 

Hopefully that connects the dots between what you heard in class, and how to figure this out on your own!