SAT II Math: Getting (RE)acquainted with Numbers
Like anything else in life, math builds on information you already know. While many of the things we need to know we really did learn in kindergarten, it's a safe bet that most of us were not taught quadratic equations and trigonometric functions in between show-and-tell and nap time in Ms. Marm's preschool classes. Just as reading and writing build on the A-B-Cs, you may need to review your 1-2-3s before tackling some of the more complex workings of math.
In fact, about 10 to 14 percent of both SAT II Math (Level IC and IIC) tests cover topics related to numbers and operations. So you want to know, for example, the difference between natural numbers and whole numbers before launching into some of the more basic problems. Otherwise, you may do all the calculations exactly right for some problem, but you could still end up with a completely wrong result if, for example, you used whole numbers when the question referred to integers. This will set you back in trying to get the best score you possibly can. Some students may end up kicking themselves for missing clues as to just what is being asked in problems that should be relatively simple.
Here are the more common types of numbers that mathematicians and real people deal with every day.
Figures you can count on: Natural numbers
Where the cave man made notches on bones to note the passing of the days in the month, the modern day kindergartner counts on her fingers. The natural numbers are those numbers starting with 1, 2, 3, 4, 5, and so on. Natural numbers are also known as counting numbers because in counting, we begin with the number 1 and continue in a series. (0 is not a counting number, naturally!) Natural numbers can also be referred to as positive integers. Wouldn't it be great if everything else were as easy as 1, 2, 3?
Add the zero: Whole numbers
Whole numbers are a whole lot like natural numbers, but they also include the number 0. In other words, whole numbers are all numbers in the following series: 0, 1, 2, 3, 4, 5, and so on. Whole numbers can also be referred to as non-negative integers. Remember that 0 is neither positive nor negative, but it is one of the whole numbers.
Numbers with a little integrity: Integers
Integers belong to the set of all positive and negative whole numbers, and they also include the number 0. Integers are not fractions or decimals or portions of a number. They really have it all together, and that's what gives them their integrity. Integers can be counted as . . . –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5. . . . Integers greater than 0 are called natural numbers or positive integers. Integers less than 0 are called negative integers. Remember that 0 is neither positive nor negative.
Splitting hairs: Rational numbers
A rational number can be expressed as the ratio of one integer to another; that is, a number that can be expressed as a fraction. Rational numbers behave rationally. Rational numbers include all positive and negative integers, plus fractions and decimal numbers that either end or repeat. For example, the fraction 1/3 can be expressed as 0.33333. . . . Rational numbers do not include numbers such as π or a radical such as √2, because such numbers cannot be expressed as fractions consisting of only two integers.
Having it all: Real numbers
Real numbers cast the widest net of all. They include all numbers that we normally think of and deal with in everyday life. For real! Real numbers belong to the set that includes all whole numbers, fractions, and rational as well as irrational numbers. Think of real numbers as those numbers represented by all the points on a number line, either positive or negative. Also think of real numbers as those numbers that you can use to measure length, volume, or weight.
In fact, it's hard to imagine a number that is not a real number, because if a number weren't real, it would be imaginary. We simply assume, when we refer to any number, that it is a real number without stating it in so many words. If you are ever asked on the SAT II Math test to give an answer expressed in terms of real numbers, you can probably guess that your answer should be the number you were planning on choosing anyway. Don't be taken in by the gratuitous red herring when they throw in million-dollar words like real number. It's more information than you probably need to solve the problem.
A league of their own: Prime numbers
Prime numbers are all positive integers that can be divided only by themselves and 1. The number 1 is not a prime number. The smallest prime number is 2, and it is also the only even prime number. This does not mean, however, that all odd numbers are prime numbers. 0 can never be a prime number either, because you could divide 0 by every natural number there is you would still come out with 0. To get the prime numbers, just think of a series of numbers beginning with 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and so on. What makes them so special is that the only two factors for these numbers will always be the number 1 and the prime number.
Not to confuse things any further, but a composite number is any natural number that is not a prime number, and it does not include the number 1. In other words, a composite number is composed of more than two factors. It is the product of more than simply itself and the number 1.
Which of the following expresses 90 as a product of prime numbers?
- 2 × 2 × 3 × 5
- 2 × 2 × 2 × 15
- 2 × 3 × 3 × 5
- 2 × 3 × 5
- 1 × 2 × 5 × 9
This question tests your knowledge of prime numbers. Remember that prime numbers are those numbers you can divide by 1 and the value of the number. (The first prime number is 2.) And remember that 1 and 0 are not prime numbers. You can easily eliminate a couple of answers, B and E, because 15 and 9 are not prime numbers. Also, E has the number 1, which is also not a prime number. So cross out B and E.
The product of A is 60, so that's not right. The product of the numbers in D is even less, 30, so that can't be right either. C is the correct answer; it contains the only numbers that are prime, and they equal 90 when you multiply them together.
It never ends: Irrational numbers
Just like it sounds, an irrational number is any real number that is not rational. Some help, eh? Just think of the definition of rational number, and realize that an irrational number is one that can't be expressed as a fraction or ratio of one integer to another. Irrational numbers are numbers such as π or any radical such as √2 that cannot be simplified any further. An irrational number, if expressed as a decimal, will go on forever without repeating itself.
Not all there: Imaginary numbers
An imaginary number, just like it sounds, is any number that is not a real number. Are you getting a kick out of this circular reasoning?
Suffice it to say that an imaginary number is a number such as √-2. As you know, any real number, whether positive or negative, when multiplied by itself (squared) results in a positive number. So you can't find the square root of a negative number unless it's simply not a real number. So, an imaginary number is the square root of any negative number, or any number containing the number i, which represents the square root of –1.