What’s wrong with the fraction 1000/2000? Well nothing technically or mathematically speaking however it’s not a simple number- why not just call it 1 / 2 it would save the extra effort of writing all those zeros. In mathematics we always want to express numbers as simple as possible hence the reason we simplify and reduce fractions. Sometimes numbers can actually be in an improper form- i.e. mathematically they are not allowed.

One of the most common examples of this in algebra is irrational numbers in the denominator of a fraction (see examples below). Recall an improper number is a number that can’t be expressed as a fraction made up of integers.

The square roots of numbers that end up with decimal answers that go on and on is a common form of irrational numbers students see. There is nothing wrong with irrational numbers when they are hanging out by themselves the problem occurs when we find them in the denominators of fractions. You see we can’t divide a number by another number that’s “irrational”. How do you divide up a pizza in x irrational parts? Well the answer is we can’t so fractions that have irrational numbers in the denominator need to be addressed. The process of fixing irrational numbers in the denominator is called “rationalizing”. I’m not going to explain the entire process but I will include a few examples below. If you want to learn more about this important skill here is a link to a lesson.