Number Terminology Study Guide
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 RATIO. 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 prealgebra, 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 ThoughtNunmber 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 twodigit 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 placevalue system, and with the placevalue system, you can efficiently write numbers as large as you like.
In the placevalue 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.
How to Count Any Number of Things, No Matter How Large, With Digits
In any number, when you add on a digit, the original number grows by a factor of ten. It works like this:
 The 9 in 9,321 signifies 9 thousand.
 The 9 in 94,321 signifies 90 thousand.
 The 9 in 954,321 signifies 900 thousand.
 The 9 in 9,654,543 signifies 9 million.
 The 9 in 97,654,321 signifies 90 million.
But there is an even more powerful way to move to larger numbers. Instead of increasing by tens, you can increase by factors of a thousand. You can move forward by groups of three digits, called periods.
Fuel for ThoughtA period is a group of three digits occurring in a number that represents 1,000 or larger. Here are the periods: units thousands millions billions trillions quadrillions quintillions sextillions septillions octillions nonillions 53 thousand is written as 53,000. 53 million is written as 53,000,000. 53 billion is written as 53,000,000,000. 53 trillion is written as 53,000,000,000,000. 53 quadrillion is written as 53,000,000,000,000,000. 53 quintillion is written as 53,000,000,000,000,000,000. 53 sextillion is written as 53,000,000,000,000,000,000,000. 53 septillion is written as 53,000,000,000,000,000,000,000,000. 53 octillion is written as 53,000,000,000,000,000,000,000,000,000. 53 nonillion is written as 53,000,000,000,000,000,000,000,000,000,000. 
Let's look at the number 557,987,654,321. What place value does each period represent?
 The first (rightmost) group of three digits, 321, signifies 321 units.
 The second group of three digits, 654, signifies 654 thousand.
 The third group of three digits, 987, signifies 987 million.
 The fourth group of three digits, 557, signifies 557 billion.
Okay, you get the point. The whole number 557,987,654,321 is read as 557 billion, 987 million, 654 thousand, 321. The placevalue system solves the problem of writing large numbers.
To Know Addition is to Love Addition
Addition is simply the totaling of a column (or columns) of numbers. Addition answers the following question: How much is this number plus that number? You can use addition to figure out how many DVDs you own, or how much money you spent on those DVDs.
Example
Oliver has 61 graphic novels and Jennifer has 52 graphic novels. Oliver's sister Lena has an additional 37 graphic novels. They decide to store all their graphic novels on Oliver's bookshelf. How many graphic novels are on the bookshelf?
Start by adding the columns of numbers.
Hopefully, you figured out that they have put 150 graphic novels on Oliver's bookshelf.
Inside TrackHow to check your answersWhen you add columns of numbers, how do you know if you came up with the right answer? One way to check, or proof, your answer is to add the figures from the bottom to the top. In the example you just saw, start with 7 + 2 in the right (ones) column and work your way up. Then, carry the 1 into the second column and say 1 + 3 + 5 + 6 and work your way up again. Your answer should still come out to 150. 
SingleDigit Adding Facts
One important step to mastering addition is to know how to flash on (that is, know by heart) the singledigit adding facts, like 5 + 2 = 7 or 6 + 8 = 14.
In turn, the way to learn to flash on these facts is first to learn how to figure them out. If you don't flash on the singledigit adding facts, you will not be able to quickly and accurately add a column of multidigit numbers. You might always feel a little uneasy around numbers.
On the other hand, if you have learned to flash on the singledigit adding facts, you will be able to manage any adding problem easily. Following is a chart of the singledigit adding facts. Once you are familiar with these facts, you will be able to add more comfortably.
Inside TrackWhen working with singledigit addition facts, there are several rules to keep in mind. Zero plus a whole number always equals that number.
Adding one to a whole number moves you to the next whole number.
Two plus a number is like counting twice from that number. You will always end up on the next odd or even number, depending on whether the original number was odd or even.
It is sometimes easier to add using tens, when possible. For example, when adding 9 plus a number, bump the 9 up one, to 10, and bump the other number down one.
When adding 8 plus a number, bump the 8 up two, to 10, and bump the other number down two.
Whenever it is convenient to bump one number up to 10 and the other number down, do it.

You know that 5 plus 5 equals 10. You can use this basic addition fact to add other numbers by learning what 6, 7, 8, and 9 are in terms of 5 plus how many. No matter how long it takes, memorize these four facts:
 6 = 5 + 1
 7 = 5 + 2
 8 = 5 + 3
 9 = 5 + 4
Knowing these four facts, you have now conquered the whole lower right part of the addition table. It's that easy.
Example
 6 + 8 = ?
Think of 6 as 5 + 1 and think of 8 as 5 + 3. Now, you have (5 + 1) + (5 + 3). Rearrange the terms to group the 5s together: 5 + 5 = 10 and 1 + 3 = 4; 10 + 4 = 14, so 6 + 8 = 14.
Adding Columns of Digits
Add this column of digits:
If you flash on all your singledigit adding facts, then you can add this column in order, quickly, as follows: 8, 13 (5 + 8), 15 (13 + 2), 22 (15 + 7), 27 (22 + 5), 33 (27 + 6), 35 (33 + 2), 39 (35 + 4), 40 (39 + 1), 46 (40 + 6). As a check, add the numbers from the bottom up: 6, 7, 11, 13, 19, 24, 31, 33, 38, 46.
But suppose you don't know how to flash. Suppose that, like many people, the only addition facts you know well are the ones that add to ten, like 8 + 2 = 10 and 6 + 4 = 10. Then, you can use a trick to help you add better—look for combinations of tens. Tens are easy to add. In our example, seeing the 8 at the top of the list, you look for and find a 2. That makes your first ten. There are two 5s, and that makes for another ten, so you have 10 + 10 = 20. You will also find ten in 7 + 2 + 1 and another ten in 6 + 4. That makes 40, and there is a 6 left over, for a grand total of 46.
Adding MultiDigit Numbers
Let's say you've been asked to add 42 and 56. How do you do this? You could start at 42 and count 56 times. A different way is to start at 56, and then you have to count only 42 times. However, both these methods would take a long time.
Fortunately, there is a faster way—using the placevalue system. Look at the number 42 and you see 4 tens and 2 ones. In 56, you'll see 5 tens and 6 ones.
Using the placevalue system, you can add 4 tens and 5 tens to get 9 tens, and 2 units and 6 units to get 8 units; 9 tens and 8 units is written as 98.
Okay, now how much is 57 + 85? This example has a little wrinkle that the first one doesn't have. You'll have to "carry."
Write the numbers one under the other, as before. Then, you have the units (7 + 5 = 12) and the tens digits (5 + 8 = 13).
When the sum of a column gets to be ten or more than ten, you need to carry. In the example above, the 7 and the 5 in the units digits result in 12. We write down the 2 and carry the 1 over to the next column, where the digit 1 signifies 1 ten.
If the numbers in the hundreds column add to a twodigit number like 12, the 2 signifies 2 hundreds and the 1 signifies 10 hundreds, which is 1 thousand. (The thousands column is right there, the column to the left of the hundreds column.)
Now try your hand at adding three columns of numbers.
Did you get the correct answer? You'll know for sure if you proofed it. If you haven't, then go back right now and check your work.
Pace YourselfHow to "write" a digit on one handLet each finger represent one, and your thumb represent five. Then, you can represent the ten digits like this:

Did you get it right? Did you come up with 3,895? If you still feel a little rusty, then what you need is more practice, and the following exercises will help.
Tackling Subtraction
Subtraction is the mathematical opposite of addition. Instead of combining one number with another, you take one away from another.
Let's start off by working out some basic subtraction problems. These problems are simple because you don't have to borrow or cancel any numbers. Try working out these problems on your own before you scroll down to the answers.
Examples
Fuel for ThoughtThe result of subtracting 5 from 8, which is 3, is called the difference between 8 and 5. When you calculate 8 minus 5, the 8 is called the minuend and the 5 is called the subtrahend. 
Inside TrackThere's a great way to check or proof your answers. Just add your answer to the number you subtracted and see if they add up to the number you subtracted from. For the previous examples, does 15 + 53 = 68? Does 41 + 36 = 77? Does 53 + 41 = 94? 
MultiDigit Subtraction
Now, I'll add a wrinkle. You're going to need to "borrow." Are you ready?
Example
You need to subtract 9 from 4. Well, that's pretty hard to do. So, you need to make the 4 into 14 by borrowing 1 from the 5 of 54. What you're doing here is converting one ten (a 1 in the tens place) to ten ones. Okay, so 14 – 9 is 5. Because you borrowed 1 from the 5, the 5 is now 4. And 4 – 4 is 0. So, 54 – 49 = 5.
Now, we'll take subtraction one step further. Do you know how to do the twostep dance? If not, no worries. Now you're going to be doing the subtraction threestep, or at least subtracting with threedigit numbers.
Examples
Solutions
Welcome to the Stage … Multiplication!
Multiplication is really just another form of addition. For instance, how much is 5 × 4? You might know it's 20 because you searched your memory for that multiplication fact. There's nothing wrong with that. As long as you can remember what the answer is from the multiplication table, you're all right.
Another way to calculate 5 × 4 is to add them: 4 + 4 + 4 + 4 + 4 = 20.
You use multiplication in place of addition, because it's shorter. Suppose you had to multiply 395 × 438. If you set this up as an addition problem, you'd be working at it for a couple of hours.
Do you know the multiplication table? You might know most of these facts by heart, but many people have become so dependent on their calculators that they've forgotten a few multiplication problems—like 9 × 6 or 8 × 7.
Multiplication is basic in understanding math. And to really know how to multiply, you need to know the entire multiplication table by memory.
Test yourself. First, fill in the answers to the multiplication problems in the table that follows. Then check your work against the numbers shown in the completed multiplication table on page 38. If they match, then you have a good grasp of your multiplication facts. But if you missed a few, then you need to practice those until you've committed them to memory.
Fuel for Thought8 × 5 is the same as 5 × 8Here is a diagram of 8 rows, each one having 5 bagels.
If you look at the diagram differently, you can see 5 columns, each one having 8 bagels. These two ways of looking at the diagram demonstrate that 8 × 5 is the same as 5 × 8. This concept is true of all the multiplying facts. 
Let's look into the realm of long multiplication. Long multiplication is just multiplication combined with addition.
Example
First, multiply 6 by 7, which gives us 42. Write down the 2 and carry the 4:
Now, multiply 4 by 7, which gives you 28. Add the 4 you carried to 28 and write down 32:
Next, multiply 6 by 3, which should yield 18. Write down the 8 and carry the 1:
Now, calculate 4 × 3, giving you 12. Add the 1 you carried to 12, and write down 13:
Then, you can add your columns:
Inside TrackTo prove your multiplication, just reverse the numbers you are multiplying. For the preceding example, check your work this way: You should still end up with the same final product. 
Try another problem just to test your skills.
Example
Well, 7 × 9 = 63. Write down the 3 and carry the 6.
You know that 7 × 8 = 56 and 56 + 6 = 62. Write down 62.
Next, 5 × 9 = 45, so write down the 5 and carry the 4.
Then, 5 × 8 = 40 and 40 + 4 = 44. Write down 44. Then, add the two columns.
Pace YourselfFlash CardsA good way to drill basic multiplication facts is to create flash cards. Using 3by5 index cards, you can create a useful set of flash cards for studying on the go. For example, write "6 × 8" and "8 × 6" on one side of an index card, and "48" on the other side. You can study either side of the flash card and try to say what's on the other side. This will remind you that you can get a product, like 48, by multiplying different pairs of numbers. 
Inside TrackHere are some singledigit multiplication facts: Zero times any number is zero.
One times any number is that number.
Two times any number is that number plus itself.
Three times any number is the number plus the double of the number.
Four times any number can be calculated by doubling the number, and then doubling that result (because 4 = 2 × 2). To calculate 4 × 7, double the 7 (14) and then double the 14 (28). Five times any even number is like taking half the number and then multiplying by ten. For 5 × 8, take half of 8 (4) and multiply by 10 (40). Six times any number can be calculated by tripling the number and then doubling that result (because 6 = 3 × 2). For 6 × 8, triple the 8 (24), and then double 24 (48). Eight times any number can be calculated by doubling the number, and then doubling that result, and then doubling that last result once again. To calculate 8 × 6, double the 6 (12), then double the 12 (24), and then double the 24 (48). Nine times any number can be calculated by tripling the number and then tripling that result. For 9 × 9, triple 9 (27) and then triple 27 (81). 
And Finally … Division
As you'll see, division is the opposite of multiplication. So you really must know the multiplication table to do division problems. Let's look at a short division problem.
You can divide 7 into 21 to get 3.
Then, try to divide 7 into 1. Because 7 is larger than 1, it doesn't fit. So we write 0 over the 1:
How many times does 7 go into 14?
You can check your answer using multiplication: 302 × 7 = 2,114.
This example came out even, but sometimes, there's a remainder. That's the case in the examples that follow.
Examples
Long Division
Long division is carried out in two steps:
 trial and error
 multiplication
The process of long division is identical to short division, but it involves a lot more calculation. That's why it's so important to have memorized the multiplication table.
Example
How many times does 37 go into 59? Just once. So, put a 1 directly above the 9 and write 37 directly below 59.
Then, you subtract 37 from 59, leaving you with 22. Next, bring down the 6, giving you 226. How many times does 37 go into 226? You need to do this by trial and error. You finally come up with 6, because 6 × 37 = 222.
Fuel for ThoughtThe dividend is the number being divided; the divisor is the number that divides the dividend. The quotient is the result. In the division problem 12 ÷ 4, 12 is the dividend and 4 is the divisor. The quotient is 3. In the division problem 13 ÷ 4, 3 is the quotient and there is a remainder of 1. 
When you subtract 222 from 226, you are left with 4, which is the remainder.
The proper notation for the answer is 16 R4. To check this answer, multiply 16 and 37 and add 4. Did you get 596?
A Glimpse of Prime Numbers
Prime numbers are whole numbers greater than 1, whose only factors are 1 and themselves.
You can easily identify prime numbers. For example, look at the number 10. The factors of 10 are 2 and 5. So, 10 is not prime. Ten is a composite number, because it is "composed" (in a multiplicative sense) of 2 and 5.
On the other hand, 11 is a prime number. Its only factors are 1 and 11.
Every whole number greater than 1 is either prime or composite. Here are the first primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67.
Fuel for ThoughtNotice that there are some pairs of primes that differ by only 2, like 11 and 13 or 59 and 61. These pairs of primes are called twin primes. 
How Do You Factor Whole Numbers?
Every whole number (except 0 and 1) is either prime or the product of primes. For example, 11 is a prime and 12 is the product of primes (2 × 2 × 3 = 12).
Inside TrackDivisibility rules for 2, 3, 5, and 7Divisibility rule for 2: Look only at the units digit. If the units digit of the number is 0, 2, 4, 6, or 8, then the number is even, i.e., the number is divisible by 2. The number 35,976 is even because its unit digit is 6. Divisibility rule for 3: Look at all the digits. If the sum of all the digits is divisible by 3, then the number is divisible by 3. For instance, 63,156 is divisible by 3 because the sum of its digits, which is 21, is divisible by 3. Divisibility rule for 5: Look only at the units digit. If the units digit of the number is 0 or 5, then the number is divisible by 5. For example, 745,980 is divisible by 5 because its units digit is 0. Divisibility rule for 7: Look at the whole number. If the number is less than 70, see if you recognize the number as a multiple of 7 (7, 14, 21, 28, 35, 42, 49, 56, 63, 70). For a twodigit number like 91, subtract out 70 and see whether the result (91 – 70 = 21) is a multiple of 7; 91 is divisible by 7 because when you subtract 70 from 91, you get 21, which is a multiple of 7. 
How can you factor any whole number less than 25? Test for divisibility by 2 and by 3. If the number that is less than 25 is not divisible by 2 and it is not divisible by 3, then it is prime.
Example
Factor 12 into its prime factors. Is 12 divisible by 2? Yes, 12 ÷ 2 = 6. Write down the factor 2. Is 6 divisible by 2? Yes, 6 ÷ 2 = 3. Write down the factor 2 again; 3 is prime. You now know that 12 = 2 × 2 × 3.
Fuel for ThoughtWhen you write 3 × 5 = 15, the 3 and the 5 are factors and the 15 is the product. The word factor works as a verb as well as a noun. When you figure out that the factors of 15 are 5 and 3, you are factoring 
Example
Factor 13 into its prime factors. Is 13 divisible by 2? No. Is 13 divisible by 3? No. Then, remember, it's prime.
Fuel for ThoughtThe number 15 is divisible by 3 because 15 has no number when divided by 3; 3 goes into 15 evenly. A number like 12 has 2 as a factor is divisible by 2. Such a number is an even number. A number like 13 that leaves a remainder when divided by 2 is an odd number. 
How can you factor any whole number less than 49? Test for divisibility by 2, 3, and 5. If the number is not divisible by 2, 3, or 5, it is prime.
Examples
Factor 21 into its prime factors. Is 21 divisible by 2? No. Is 21 divisible by 3? Yes. Write down the 3: 21 ÷ 3 = 7; 7 is prime, so the factors of 21 are 3 and 7.
Factor 43 into its prime factors. Is 43 divisible by 2? No. Is 43 divisible by 3? No. Is 43 divisible by 5? No, so 43 is a prime number.
How can you factor any whole number less than 121? Test for divisibility by 2, 3, 5, and 7. If the number is not divisible by 2, 3, 5, or 7, then it is prime.
Example
Factor 89 into its prime factors. Is 89 divisible by 2? No. Is 89 divisible by 3? Again, no. Is 89 divisible by 5? No; almost done. Is 89 divisible by 7? No—okay, so 89 is a prime number.
Find practice problems and solutions for these concepts at Number Terminology Practice Problems.

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