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What Are Irrational Numbers?

Updated on July 02, 2016
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Leonard Kelley holds a Bachelor's in Physics with a minor in Mathematics. He loves the academic world and strives to constantly improve it.

To better understand irrational numbers, we need to know what a rational number is and the distinction it has from an irrational number. This is simply a number that can be defined as a fraction of two whole, or non-decimal, numbers. 5 is rational because it can be expressed as the fraction 5/1 which equals 5. 1.6 is also rational because 16/10 = 1.6. Irrational numbers are the opposite of rational numbers: They cannot be expressed by a fraction involving two whole numbers, no matter how large you make them. The best you can do is to write out the number as a non-repeating fraction or decimal, which will go on and on for forever. They include the following:


When we use powers, we are indicating how many times we are multiplying a number. Some examples include:

22 = 2*2 = 4

53 = 5*5*5 = 125

13 = 1*1*1 = 1

Some care must be taken regarding powers. As you can see from the prior examples, some are rational. So when would a power make the result an irrational number? Let's look at this example:

41/2 = Square Root of 4 = 2

is a whole number (2/1). However, the same cannot be said for


because that is roughly 1.4 after rounding. Since rounding was involved, the actual solution is not a fraction of two whole numbers. It would continue on as a decimal forever, never-ending. Another example is


which equals 5.2 roughly. As we can see, powers that result in irrational numbers are often reliant on the number it is raising.


This is the ratio of a circle's circumference to it's diameter, roughly 3.14. However, no one has yet been able to fully solve what that ratio actually equals, but it has been solved to a very extensive point. Below is Pi solved to a few thousand decimal places.

Some properties of logarithms.
Some properties of logarithms. | Source


This is the process for determining what power I raise a number to for a given result. Generally,

Log10(x) = y or 10y = x

For example

Log10(1) = 0

which means that 10 raised to the 0 power would equal one (100 = 1). However, you will come across irrational values such as

Log10(2) = 0.301 approximately.

That is, 100.301 = 2 approximately.

These are but a sampling of all the other irrational numbers that exist. Numbers involving trigonometry (cosines sine, tangents, etc.), natural ratios (golden ratio) and everything presented here have the capacity to be an irrational number. An infinite number of them are out there, so finding them is not as difficult as it may seem. They are everywhere we look and frequently where we least expect it.

© 2009 Leonard Kelley


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      Rod 4 years ago

      I think this statement is a mistake: 5/2 = 2.5 being irrational

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      Leonard Kelley 4 years ago

      You are indeed correct, I need to fix that. Thank you!

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