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What are Numbers?

Numbers represent amounts of things. In math, we keep things very abstract and vague. When your teacher asks you the answer of 2+2, you say 4. We aren't really talking about anything in particular. Numbers at that level are just an idea or concept. When you get into the real world, numbers represent real things. If you have 2 apples and get another 2, you have 4 apples. Numbers can tell you how many apples you have.

The western system of numerals uses a base ten system. It is based on a long history of math going back to ancient India. You will see it all the time when you work with decimals. We have ten digits (0 through 9) that we move around and combine to represent new values. When you just have a single digit, it represents that exact number of things (for example, a 5 equals five things). Once you move past nine, you add a second digit in front of the first that represents ten things. Fifteen (15) is a representation of ten things and five things. It keeps moving up from there into hundreds, thousands, tens of thousands...and it just keeps going. You know all of this.

• 0 = no things.
• 5 = five things.
• 15 = one ten of things and five things.
• 145 = one hundred things, four tens of things, and five things.

There have been other types of numeral systems, but they aren't used anymore. One of the most famous was the Roman numeral system, where they used letters that include I, V, X, L, C, and M. A number such as two hundred fifty-five (255) would have been written CCLV. They understood the idea of zero, but did not have a symbol that represented the value.

Real Numbers

If you haven't figured it out by now, mathematicians like to keep things organized. Our numbers go in just the right places. We use commas and points to separate values. Arithmetic symbols have different and specific actions. We also organize our numbers into groups. The biggest group of numbers is called real numbers. You can find one real number for every place on a number line. Anything on that number line that we have been using — any point at all — is a real number. It is the biggest grouping of numbers we will talk about in our pages. Every number you find on that line is a real number.

Examples of Real Numbers:
• 0, 15, 8,000,000, 5/16, 321/895, π, √2
• -15, -8,000,000, -5/16, -321/895, -π, -√2
• ...any number on the number line in any direction.

Sets

We'll be talking about ideas such as sets and subsets as you move into more difficult math. We want to introduce you to the way that sets are displayed in math. A set of numbers is shown in brackets like this: { }. If the set is too big to write, you will see an ellipsis (...) to tell you that the list of numbers just keeps going.

• {1, 3, 5} is a set of three numbers. The only numbers in the set are 1, 3, and 5.

• {1, 2, 3, 4, 5, ...} is a set of all positive natural numbers. The "..." (ellipsis) means the list keeps going forever in the positive direction.


• {... -7, -5, -3, -1, 1, 3, 5, 7, ...} is a set of all odd numbers. There are two "...", because the list goes on forever in both the positive and negative directions of the number line.


Don't worry about writing sets of numbers right now. We just wanted to introduce you to the way that they are written.

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