Number Terminology Study Guide

Updated on Aug 24, 2011

Find practice problems and solutions for these concepts at Number Terminology Practice Problems.

What's Around The Bend

  • Counting Numbers
  • Integers
  • Rational and Real Numbers
  • Place Value
  • Addition
  • Subtraction
  • Multiplication
  • Division
  • Prime Numbers
  • Factoring Whole Numbers

Numbers, numbers, numbers. Before we begin to explore mathematical concepts and properties, let's discuss number terminology. The counting numbers, 0, 1, 2, 3, 4, 5, 6, and so on, are also known as the whole numbers. No fractions or decimals are allowed in the world of whole numbers. What a wonderful world, you say. No pesky fractions and bothersome decimals.

But, as we leave the tranquil world of whole numbers and enter into the realm of integers, we are still free of fractions and decimals, but subjected to the negative counterparts of all those whole numbers that we hold so dear. The set of integers would be {… –3, –2, –1, 0, 1, 2, 3 … }.

The real numbers include any number that you can think of that is not imaginary. You may have seen the imaginary number i, or maybe you haven't (maybe you only thought you did). The point is, you don't have to worry about it. Just know that imaginary numbers are not allowed in the set of real numbers. No pink elephants either! Numbers that are included in the real numbers are fractions, decimals, zero, π, negatives, and positives of many varieties.

So where do irrational and rational numbers fit into all this? Here's how it works. The first five letters of RATIONAL are R-A-T-I-O. Rational numbers can be represented as a ratio of two integers. In other words, a rational number can be written as a decimal that either ends or repeats. Irrational numbers can't be represented as a ratio, because their decimal extensions go on and on forever without repeating. π is the most famous irrational number. Other irrational numbers are √2 and √11.

So, you probably think you know the deal with whole numbers, right? They are the numbers 0, 1, 2, 3, and so forth. Yes, you already know that they go on forever and that there is no such thing as the largest whole number.

But in order to conquer basic math and pre-algebra, you need to really understand how whole numbers work with the basic operations: addition, subtraction, multiplication, and division. And then there are the issues of place value, prime numbers, factoring, and …

Okay, ready to reintroduce yourself to whole numbers? They really are quite fascinating once you get to know them.

How to Build Large Numbers Out of Small Numbers

You can write thousands of words using only the 26 letters of the English language alphabet. Think about the words you can create with just the letters M, A, T, and H. Can you also be this clever with whole numbers?

Fuel for Thought

Nunmber is a very general term that includes whole numbers, fractions (e.g., ), decimals (e.g., 24.35), and negative numbers (e.g., –24).

Digit refers to the ten special numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The number 36 is a two-digit number.

Yes, you can be extremely creative with numbers. Using only ten basic numbers, you can write numbers as large as you like. Use these basic numbers—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9—to write at least ten different number combinations on the lines below. Make them any length you want—just be sure to fit them on the page!

Okay, now you see at least ten different combinations for the basic numbers, but there are actually infinite combinations! No, we weren't going to make you try to list them all. We aren't that cruel! (We hear your sigh of relief.)

These ten basic numbers are called digits and they are very useful. If you didn't have them, imagine how you would have to express the number of days in a week. You would have to say something like, "There are 1,111,111 days in a week." Then, maybe you would eventually develop shortcuts, like using v to mean five: "There are vii days in a week." This was the beginning of the Roman numeral system, which is widely used today for lists that are not likely to go beyond about a dozen items.

Today, you have a much better system than Roman numerals. You have the place-value system, and with the place-value system, you can efficiently write numbers as large as you like.

In the place-value system, you can indicate the number of days in a year as 365. The 5 stands for 5 days; the 6 in the second place (counting from the right) stands for 6 times 10, or 60 days; and the 3 in the third place stands for 3 times 100, or 300 days. The values of the digits get ten times bigger each time you move to the left.

When you write 365, you are adding 300, 60, and 5 (300 + 60 + 5). If you put these numbers under each other, you can clearly see that they add up to 365.

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