Wednesday, October 24, 2007

Repeating/Non-repeating decimals and Terminating and Non-terminating decimals

Rational Numbers:

The word rational refers to something reasonable, understandable, within reason. In short rational numbers are whole numbers, fractions, and decimals - the numbers we CAN understand and that we use in our daily lives.

In mathematical terms a number is rational if you can write it in a form a/b where a and b are integers, b not zero. Clearly all fractions are of that form. Terminating decimal numbers can easily be written in that form: for example 0.67 is 67/100, 3.40938 = 340938/100000 etc.
Terminates means to end. So, if you decimal ends, it is a rational number. For example, .46 is a rational number because it is terminating.

Irrational Numbers:

Rational numbers are contrasted with irrational numbers - such like Pi and square roots of non-perfect squares of numbers. In a sense you can't really understand the irrational numbers, because the name irrational itself means NOT rational, NOT reasonable, NOT understandable, NOT within reasoning powers.

Repeating decimals:

A repeating decimal, also called a recurring decimal, is a number whose decimal representation eventually becomes periodic (i.e., the same sequence of digits repeats indefinitely).

.333333333333... is a repeating decimal because 3 repeats. The (...) tells you that 3 goes on and on and that the decimal is non-terminating, which I will go over shortly. (Look below)

Non-repeating decimals:

Non-repeating decimals are decimals that do not have a repeating pattern. For example, the value for Pi is a non-repeating decimal. Its value is: 3.14159265.... You cannot predict any type of pattern for pi. It's irrational.

Terminating decimals:

Terminating decimals are decimals are decimals that END; they do not go on and on and on and on. 6.5 and 0.0000034 are examples of terminating decimals. They stop. They do not go on and on.

Non-terminating decimals:

Non-terminating decimals do not end. They go on forever. The (...) or the line over your decimals indicated that your decimals are non-terminating.

For example, 6.33333333333.... is non-terminating.

Now, how do we figure out what decimals are rational or irrational?

Irrational decimals will be BOTH non-terminating and non-repeating decimals. All other types of decimals will be rational. That is, if your decimal is not BOTH non-terminating and non-repeating, then, it is not irrational.

For example, if a number is non-terminating, but it is repeating, then it is rational. .3333... is repeating and non-terminating. It is rational because it does not BOTH non-repeating and non-terminating. It satisfies ONLY one of the rules.