The SAT Prep Black Book(79)
A number’s absolute value is the distance of that number from zero on the number line.
Example:
-4 and 4 both have an absolute value of 4. We signify the absolute value of a number with vertical lines on either side of the number:
|-4| = |4| = 4.
Squares and square roots
To square a number, multiply the number by itself.
Example:
Five squared is five times five, or 5 * 5, or 25.
To find the square root of a number, find the amount that has to be multiplied by itself in order to generate the number.
Example:
The square root of 25 is the amount that yields 25 when it’s multiplied by itself. As we just saw, that amount is 5. So the square root of 25 is 5.
When you square any number, the result is always positive. This is because a positive number times a positive number gives a positive result, and so does a negative number times a negative number.
Square roots on the SAT are always positive.
The SAT never asks about the square root of a negative number.
The SAT likes to ask about the squares of the numbers -12 through 12. Here they are:
Number Square
-12 or 12 144
-11 or 11 121
-10 or 10 100
-9 or 9 81
-8 or 8 64
-7 or 7 49
-6 or 6 36
-5 or 5 25
-4 or 4 16
-3 or 3 9
-2 or 2 4
-1 or 1 1
0 0
While I don’t recommend using a calculator on the SAT if you can help it, remember that you can always find the square root of a number very easily on a good calculator.
Fractions and rational numbers
A fraction is a special type of number that represents parts of a whole.
Fractions are written this way:
[number of parts being described in the situation]
[number of parts that the whole is divided into]
Example:
Imagine that we’re sharing a six-pack of soda cans. I really like soda, so I drink five of the cans. In this situation, I’ve had five of the six cans that make up the six-pack—I’ve had 5/6 of the six-pack.
The number above the fraction bar is called a numerator.
The number under the fraction bar is called a denominator.
When the numerator of a fraction is less than the denominator, the value of the fraction is less than 1.
When the numerator of a fraction is greater than the denominator, the value of the fraction is greater than 1.
Example:
1/2 is equal to one half, which is less than 1. 6/3 is equal to 2, which is greater than 1.
Any integer can be thought of as having the denominator 1 already underneath it.
Example:
7 is the same thing as 7/1.
A reciprocal is what you get if you switch the numerator and the denominator of a fraction.
Example:
The reciprocal of 2/3 is 3/2. The reciprocal of 7 is 1/7. (Remember that all integers can be thought of as having the denominator 1.)
To multiply two fractions, first multiply their numerators and write that amount as the numerator of the new fraction; then, multiply the denominators and write that amount as the denominator of the new fraction.
Example:
4/7 x 9/13 = 36/91
To divide fraction a by fraction b, we actually multiply fraction a by the RECIPROCAL of fraction b.
Example
4/7 divided by 9/13 = 4/7 x 13/9 = 52/63
Multiplying a non-zero integer by a fraction that’s less than 1 (that is, by a fraction where the numerator is less than the denominator) will give a result that is closer to zero on a number line than the original integer was. (Read this item again if you need to!)
Examples:
6 x 3/5 = 18/5, and 18/5 falls between 0 and 6 on a number line.
-7 x 2/9 = -14/9, and -14/9 falls between -7 and 0 on a number line.
Fraction a is equal to fraction b if you could multiply the numerator in a by a certain number to get the numerator in b, and you could also multiply the denominator in a by the same number to get the denominator of b.
Example:
3/5 is equal to 18/30 because 3 x 6 = 18 and 5 x 6 = 30. Here’s another way to write this: 3/5 x 6/6 = 18/30. Notice that 6/6 is the same thing as 1 (six parts of a whole that’s divided into six parts is the same thing as the whole itself). So all we really did here was multiply 3/5 by 1, and we know that doing this will give us an amount equal to 3/5.
For more on fractions, see the discussion of factors and multiples below.
Factors
The factors of a number x are the positive integers that can be multiplied by each other to achieve that number x.
Example:
The number 10 has the factors 5 and 2, because 5 * 2 = 10. It also has the factors 10 and 1, because 1 * 10 = 10.
“Common factors,” as the name suggests, are factors that two numbers have in common.
Example:
The number 10 has the factors 1, 2, 5, and 10, as we just saw. The number 28 has the factors 1, 2, 4, 7, 14, and 28. So the common factors of 10 and 28 are 1 and 2, because both 1 and 2 can be multiplied by positive integers to get both 10 and 28.
Multiples
The multiples of a number x are the numbers you get when you multiply x by 1, 2, 3, 4, 5, and so on.