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      The square root of a negative number is called an imaginary number — that is, a real number multiplied by .

      Only a few SAT questions include imaginary numbers. You learn more about imaginary numbers in Chapter 12, where I focus on the roots of quadratic functions, and in Chapter 15, where I discuss operations with imaginary numbers.

      Complex numbers

      The complex numbers are the set of all numbers of the form , where a and b are both real numbers and . Another way to think of this is that a is a real number and bi is an imaginary number.

      The complex numbers include the set of real numbers, the set of imaginary numbers, and other values. Like the rational numbers and real numbers, the complex numbers are closed under the basic four operations. They’re also closed under square roots and a variety of other operations.

      Very few questions on the SAT require knowledge of the complex numbers. I discuss the specific points you need to know about them in Chapters 12 and 15.

Fractions, Ratios, Decimals, and Percentages

      Fractions, ratios, decimals, and percentages are four complementary ways of describing rational numbers — that is, the values that lie between the integers on the number line. In this section, you get a quick review of how to work with these important mathematical values.

      Review of fractions and ratios

      A fraction is composed of two integers: a numerator (top number) divided by a denominator (bottom number). For example:

      The reciprocal (or inverse) of a fraction is the result when you exchange the numerator and denominator. For example:

      Converting between improper fractions and mixed numbers

      A proper fraction has a numerator that’s less than its denominator. In contrast, an improper fraction has a numerator that’s greater than or equal to its denominator.

      Improper fractions can be awkward, because in many real-world cases, an improper fraction doesn’t provide easy-to-understand numerical information. For example, if I tell you that I bought gallons of gasoline, you may have a hard time interpreting that information.

However, if I convert the improper fraction to its mixed number of , you now know that I bought a little more than 9 gallons.

       To change an improper fraction to a mixed number, divide the numerator by the denominator. If the result has a remainder, use the remainder as the numerator of the answer. For example, to convert to a mixed number, divide , so .

      To convert a mixed number to an improper fraction, multiply the denominator by the whole number, add the numerator, and then use this number as the numerator of the answer. For example, to convert to an improper fraction, calculate , so .

      Finding simplified and increased forms of fractions

      Sometimes when a fraction has a large numerator and denominator, you can simplify it by dividing both of these numbers by the same value, resulting in an equivalent fraction. For example:

      The most simplified form of a fraction is usually the form that an SAT answer will take.

      You can also reverse this process to increase the denominator of a fraction by multiplying the numerator and denominator by the same value.

      Increasing the denominator of a fraction is often useful for adding and subtracting fractions, which I discuss in the next section.

      Adding and subtracting fractions

      When a pair of fractions both have the same denominator, you can add or subtract them by adding or subtracting their numerators and keeping the denominator the same.

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