Irrational numbers

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Remember when I wrote that decimals were another way of writing fractions?
I lied. There are some decimals that could never be changed into fractions.

Now you have already seen that any finite decimal can be transformed into a fraction.
Examples: 0.345, 2.14, and 1.34765

You have also seen infinite repeating decimals changed into fractions.
Examples: 0.33333…, 2.14567567567…, and 3.456712444444…

All of these numbers are rational numbers.
A rational number is a number that can be expressed as a fraction.
All of the numbers we have seen throughout the entire tutorial thus far are rational numbers.
The opposite of rational numbers is irrational numbers.

An irrational number is one which cannot be expressed as a fraction.
Examples of irrational numbers:

Notice that none of the above three decimals is finite, nor are any of them infinite repeating decimals.

Now I'm going to present you with 6 examples in which we will determine whether a number is rational or irrational

Is "0.83333333333333333333333333333333…" a rational or an irrational number?

In this decimal the digit "3" is infinitely repeated.
Therefore it can be transformed into a fraction.
In fact it is equal to the fraction 5/6.
Thus "0.83333333333333333333333333333333…" is a rational number.

Is "3.60555127546398929311922126747050…" a rational or an irrational number?

This is an infinite decimal, and it does not appear to be an infinite repeating decimal.
It is therefore cannot be transformed in a fraction, and is considered to be an irrational number.

Is "3.605551275463989293119221267470507050705070507050…" a rational or an irrational number?

This number is obscenely long, but eventually the digits "7050" start repeating over and over again.
Therefore this is a repeating decimal, which makes it a rational number.
I don't know about you, but I for one wouldn't want to even attempt changing it into a fraction.

Is "3.1415926535897932384626433832795…" a rational or an irrational number?

This is an infinite decimal, and it does not appear to be a infinite repeating decimal.
It is therefore an irrational number.
In fact this is a very important irrational number known as "pi." We will discuss it at length later in this unit.

Is "0.31415926535897932384" a rational or an irrational number?

This gargantuan number is obscenely long, but it does end.
Therefore it is a finite decimal and can be transformed into the fraction
31,415,926,535,897,932,384/100,000,000,000,000,000,000
Isn't that a sight to behold!
Nonetheless 0.31415926535897932384 is a rational number.

Is 5.6 a rational or irrational number?

5.6 is a finite decimal.
It can be changed into the fraction 5 3/5 (or the improper fraction 28/5).
5.6 is considered to be rational.

Now it's your turn!
Click the "generate a problem" button to see a number. Now decide whether it is rational or crazy (oops, I meant irrational), and click on the appropriate choice. The computer will tell you whether or not you made the correct choice.

Once you complete this problem you can generate another problem by clicking the "reset" button and then again clicking on the "generate a problem" button. Keep on practicing until you get the hang of this, and then you have he following options: