TEK 6.2 A Learning Goal: I can classify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers.

Vocabulary - Types of Numbers

Natural Numbers

Natural numbers are the numbers you most commonly use in your day-to-day life. Also known as counting numbers, natural numbers are the numbers you would use to count how many objects (pennies, buttons, books, etc.) you have. When you count, you use the numbers 1, 2, 3, and so on. Say you were counting how many pennies you have. You can't split the pennies into pieces, and you can't have a negative amount of pennies. When you count, you start at one and add one for each penny you have. All of the possible number of pennies you can have are natural numbers.

Whole Numbers

Whole numbers are exactly like natural numbers, except they include zero as well as all the counting numbers. You might hear some people say that natural numbers can include zero too, because sometimes when counting pennies you have zero pennies. Others would say you don't have zero pennies because you have nothing, so you have no quantity of pennies. Plus, you start counting at one, so one is the first counting number. It can be argued either way, but you don't have to worry about it. Just remember that natural numbers do not include zero and whole numbers do.

Integers

Integers are the same as whole numbers, except you can put a negative sign in front of the positive numbers. Integers include -3, -2, -1, 0, 1, 2, 3, and so on. Sometimes integers can be written as fractions, like 6/2, which simplifies to the integer 3. You always need to see if a fraction can simplify into an integer before classifying it. If it can't be simplified any further, it is not an integer.

Rational Numbers

A rational number is any number that can be represented by a fraction (one integer divided by another integer.) This includes all integers. For example, the integer 2 can be represented by the fraction 4/2, which simplifies to 2, and the integer 0 can be represented by 0/1, which simplifies to 0. Rational numbers can also be decimals that aren't integers, like the rational number 0.4. 0.4 can also be written as 2/5, a fraction, so it is a rational number. Not all rational decimal numbers are short and simple. Some repeat forever, such as 0.333333333.... The ellipsis (the three dots after the numbers) show that this decimal goes on forever. Even though it repeats forever, this number can also be written as 1/3.

Irrational Numbers

Irrational numbers are decimals that can not be written as a fraction. The most famous example that you have probably heard of before is pi. Pi is simplified to 3.14, but its actual value is 3.1415926535897932384626.... The decimal goes on forever with no pattern, so there is no fraction that pi can be rewritten as.

Rational vs Irrational Numbers

You may be wondering how to tell if a decimal is a rational or irrational number just from looking at it. Just follow these steps:

1. If you have seen this decimal before in class and you know it can be written as a fraction then it is a rational number.

2. If you have never seen this decimal before, see if you can find a fraction that is equal to this number. If you can, then it is a rational number.

3. If the decimal is too long or complicated to turn into a fraction, look at how many numbers it has after the decimal point. If the decimal ends at some point instead of continuing forever, then it is a rational number.

4. If the decimal continues forever, look to see if there is some sort of eventual pattern. If there is a pattern, the number is a rational number. If there is no pattern that you can see, the number is an irrational number.

For example, the number 5/7 is equal to the decimal 0.714285714285.... The first time you look at it, it may seem like a bunch of random numbers, but if you take a close look, the decimal is just 714285 repeated over and over again. So, this is a rational number.

Another example is 1/12, which is equal to 0.08333333.... Even though the decimal is technically not a pattern because it has an eight and zero before the threes, eventually the number settles into a pattern of repeating threes. As stated above, a rational number only needs to have an eventual pattern, which means that as the number keeps going you can start to see a pattern. It doesn't matter if the first couple numbers do not follow the pattern, as long as the number eventually settles into a pattern.

Putting it all Together

By now, you may have noticed that a single number can be classified as multiple number types. All natural numbers and the number zero make up the whole numbers, all whole numbers are integers, and all integers are rational numbers. We can put these relationships into a Venn diagram:

Irrational numbers would be somewhere off to the side in their own land because they don't fit into any of the other categories, and none of the other categories fit into them.

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