Math Fundamentals: Integers, Prime Numbers, Exponents, and Roots, Study Guides, Projects, Research of Mathematics

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Chapter 3

Math Fundamentals

DEALING WITH NUMBERS

Whenever you decide to learn a new language, what do they start with on the very first day? Vocabulary. Well, math has as much of its own lexicon as any country’s mother tongue, so now is as good a time as any to familiarize yourself with the terminology. These vocabulary words are rather simple to learn—or relearn—but they’re also very important. Any of the terms you’ll read about in this chapter could show up in a GRE math question, so you should know what the test is talking about. (For a more lengthy list, you can consult the glossary in Chapter 14.)

We’ll start our review with the backbone of all Arabic numerals: the digit.

Digits

You might think there are an infinite number of digits in the world, but in fact there are only ten: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This is the mathematical “alphabet” that serves as the building block from which all numbers are constructed.

Modern math uses digits in a decile system, meaning that every digit in a number represents a multiple of ten. For example, 1,423.795 = (1 × 1,000) + (4 × 100) + (2 × 10) + (3 × 1) + (7 × 0.1) + (9 × 0.01) + (5 × 0.001).

You can refer to each place as follows:

• 1 occupies the thousands place.
• 4 occupies the hundreds place.
• 2 occupies the tens place.
• 3 occupies the ones , or units , place.
• 7 occupies the tenths place, so it’s equivalent to seven tenths, or.
• 9 occupies the hundredths place, so it’s equivalent to nine hundredths, or.
• 5 occupies the thousandths place, so it’s equivalent to five thousandths, or.

When all the digits are situated to the left of the decimal place, you’ve got yourself an integer.

Integers

When we first learn about addition and subtraction, we start with integers , which are the numbers you see on a number line.

Integers and digits are not the same thing; for example, 39 is an integer that contains two digits, 3 and 9. Also, integers are not the same as whole numbers , because whole numbers are non-negative, which include zero.

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Explanations for Digits Quick Quiz

1. If x , y , and z are consecutive even integers and x < 0 and z > 0, then x must be −2, y must be 0, and z must be 2. Therefore, their product is 0, and you would enter this number into the box.
2. Take the answer choices and switch the hundreds digit and ones digit. When the result is 396 less than the old number, you have a winner. Choices (A), (B), and (C) are out, because their ones digits are greater than their hundreds digits; therefore, the result will be greater (for example, 293 becomes 392). If you rearrange 713, the result is 317, which is 396 less than 713. The answer is (D).
3. Pick three consecutive digits for a , b , and c , such as 2, 3, and 4. Quantity A becomes 2 × 3 × 4, or 24, and Quantity B becomes 2 + 3 + 4, or 9. Quantity A is greater so eliminate (B) and (C). But if a , b , and c are −1, 0, and 1, respectively, then both quantities become 0 so eliminate (A). Therefore, the answer is (D).

MORE ABOUT NUMBERS

Prime Numbers

Prime numbers are special numbers that are divisible by only two distinct factors: themselves and 1. Since neither 0 nor 1 is prime, the least prime number is 2. The rest, as you might guess, are odd, because all even numbers are divisible by two. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Note that not all odd numbers are prime; 15, for example, is not prime because it is divisible by 3 and 5. Said another way, 3 and 5 are factors of 15, because 3 and 5 divide evenly into 15. Let’s talk more about factors.

Factors

As we said, a prime number has only two distinct factors: itself and 1. But a number that isn’t prime—like 120, for example—has several factors. If you’re ever asked to list all the factors of a number, the best idea is to pair them up and work through the factors systematically, starting with 1 and itself. So, for 120, the factors are

• 1 and 120
• 2 and 60
• 3 and 40
• 4 and 30
• 5 and 24
• 6 and 20
• 8 and 15
• 10 and 12

Notice how the two numbers start out far apart (1 and 120) and gradually get closer together? When the factors can’t get any closer, you know you’re finished. The number 120 has 16 factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. Of these factors, three are prime (2, 3, and 5).

That’s also an important point: Every number has a finite number of factors.

Prime Factorization

Sometimes the best way to analyze a number is to break it down to its most fundamental parts—its prime factors. To do this, we’ll break down a number into factors, and then continue breaking down those factors until we’re stuck with a prime number. For instance, to find the prime factors of 120, we could start with the most obvious factors of 120: 12 and 10. (Although 1 and 120 are also factors of 120, because 1 isn’t prime, and no two prime numbers can be multiplied to make 1, we’ll ignore it when we find prime factors.) Now that we have 12 and 10, we can break down each of those. What two numbers can we multiply to make 12? 3 and 4 work, and since 3 is prime, we can break down 4 to 2 and 2. 10 can be broken into 2 and 5, both of which are prime. Notice how we kept breaking down each factor into smaller and smaller pieces until we were stuck with prime numbers? It doesn’t matter which factors we used, because we’ll always end up with the same prime factors: 12 = 6 × 2 = 3 × 2 × 2, or 12 = 3 × 4 = 3 × 2 × 2. So the prime factor tree for 120 could look something like this:

The prime factorization of 120 is 2 × 2 × 2 × 3 × 5, or 2^3 × 3 × 5. Note that these prime factors (2, 3, and 5) are the same ones we listed earlier.

Multiples

identified by the mnemonic device that most of us come in contact with sooner or later at school—PEMDAS, which stands for P arentheses, E xponents, M ultiplication and D ivision, and A ddition and S ubtraction. (You might have remembered this as a kid by saying “Please Excuse My Dear Aunt Sally,” which is a perfect mnemonic because it’s just weird enough not to forget. What the heck did Aunt Sally do, anyway?)

In order to simplify a mathematical term using several operations, perform the following steps:

1. Perform all operations that are in parentheses.
2. Simplify all terms that use exponents.
3. Perform all multiplication and division from left to right. Do not assume that all multiplication comes before all division, as the acronym suggests, because you could get a wrong answer. WRONG : 24 ÷ 4 × 6 = 24 ÷ (4 × 6) = 24 ÷ 24 = 1. RIGHT : 24 ÷ 4 × 6 = (24 ÷ 4) × 6 = 6 × 6 = 36.
4. Fourth, perform all addition and subtraction, also from left to right.

It’s important to remember this order, because if you don’t follow it, your results will very likely turn out wrong.

Try it out in a GRE example.

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Here’s How to Crack It Simplify (2 + 1)^3 + 7 × 2 + 7 − 3 × 4^2 like this:

Parentheses: (^) ( 3 )^3 + 7 × 2 + 7 − 3 × 4^2 Exponents: 27 + 7 × 2 + 7 − 3 × 16 Multiply and Divide: 27 + 14 + 7 – 48 Add and Subtract: 41 + 7 − 48 48 − 48 = 0

The answer is (B).

Working with Numbers Quick Quiz

The number of even multiples of 11 between 1 and 100 The number of odd multiples of 22 between 1 and 100

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The remainder when 33 is divided by 12 The remainder when 200 is divided by 7

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PARTS OF THE WHOLE (FRACTIONS AND DECIMALS)

It’s still necessary to be knowledgeable when it comes to fractions, decimals, and percents. Each of these types of numbers has an equivalent in the form of the other two, and fluency among the three of them can save you precious time on test day.

For example, say you had to figure out 25% of 280. You could take a moment to realize that the fractional equivalent

of 25% is. At this point, you might see that of 280 is 70, and your work would be done.

If nothing else, memorizing the following table will increase your math IQ and give you a head start on your calculations.

The Conversion Table

Fractions

Each fraction is made up of a numerator (the number on top) divided by a denominator (the number down below). In other words, the numerator is the part , and the denominator is the whole. By most accounts, the part is less than the whole, and that’s the way a fraction is “properly” written.

Improper Fractions

For a fraction, when the part is greater than the whole, the fraction is considered improper. The GRE won’t quiz you on the terminology, but it usually writes its multiple-choice answer choices in proper form. A proper fraction takes this form:

common denominator.” It might be a convenient thing to learn in order to impress your math teacher, but on the GRE it’s way too much work.

The Bowtie

The Bowtie method has been a staple of The Princeton Review’s materials since the company began in a living room in New York City in 1981. It’s been around so long because it works so simply.

To add and , for example, follow these three steps:

Step One: Multiply the denominators together to form the new denominator.

Step Two: Multiply the first denominator by the second numerator (5 × 4 = 20) and the second denominator by the first numerator (7 × 3 = 21) and place these numbers above the fractions, as shown below.

See? A bowtie!

Step Three: Add the products to form the new numerator.

Subtraction works the same way.

Note that with subtraction, the order of the numerators is important. The new numerator is 21 − 20, or 1. If you

somehow get your numbers reversed and use 20 − 21, your answer will be − , which is incorrect. One way to keep

your subtraction straight is to always multiply up from denominator to numerator when you use the Bowtie so the

product will end up in the right place.

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Here’s How to Crack It First, eliminate (D) because both Quantity A and Quantity B contain numbers. Next, the Bowtie can also be used to compare fractions. Multiply the denominator of the fraction in Quantity B by the numerator of the fraction in Quantity A and write the product (40) over the fraction in Quantity A. Next, multiply the denominator of the fraction in Quantity A by the numerator of the fraction in Quantity B and write the product (39) over the fraction in Quantity B. Since 40 is greater than 39, the answer is (A).

Multiplying Fractions

Multiplying fractions isn’t nearly as complicated as adding or subtracting, because any two fractions can be multiplied by each other exactly as they are. In other words, the denominators don’t have to be the same. All you have to do is multiply all the numerators to find the new numerator, and multiply all the denominators to find the new denominator, like this:

The great thing is that it doesn’t matter how many fractions you have; all you have to do is multiply across.

Knowing this will help you devise a nice shortcut for working with problems such as the following.

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Here’s How to Crack It

To solve this problem, multiply each answer choice by the original expression,. If the product is 1, you know you

have a match. In this case, the only expression that works is , so the answer is (C).

Reducing Fractions

Are you scared of reducing, or canceling, fractions because you’re not sure what the rules are? If so, there’s only one rule to remember:

You can do anything to a fraction as long as you do exactly the same thing to both the numerator and the denominator.

When you reduce a fraction, you divide both the top and bottom by the same number. If you have the fraction , for

example, you can divide both the numerator and denominator by a common factor, 3, like this:

Be Careful If you are worried about when you can cancel terms in a fraction, here’s an important rule to remember. If you have more than one term in the numerator of a fraction but only a single term in the denominator, you can’t divide into one of the terms and not the other.

The only way you can cancel something out is if you can factor out the same number from both terms in the numerator and then divide.

Decimals

Decimals are just fractions with a hidden denominator: Each place to the right of the decimal point represents a fraction.

Comparing Decimals

To compare decimals, you have to look at the decimals place by place, from left to right. As soon as the digit in a specific place of one number is greater than its counterpart in the other number, you know which is bigger.

For example, 15.345 and 15.3045 are very close in value because they have the same digits in their tens, units, and tenths places. But the hundredths digit of 15.345 is 4 and the hundredths digit of 15.304 is 0, so 15.345 is greater.

Rounding Decimals

In order to round a decimal, you have to know how many decimal places the final answer is supposed to have (which the GRE will usually specify) and then base your work on the decimal place immediately to the right. If that digit is 5 or higher, round up; if it’s 4 or lower, round down.

For example, if you had to round 56.729 to the tenths place, you’d look at the 2 in the hundredths place, see that it was less than 5, and round down to 56.7. If you rounded to the hundredths place, however, you’d consider the 9 in the thousandths place and round up to 56.73.

Fractions and Decimals Quick Quiz

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Explanations for Fractions and Decimals Quick Quiz

1. Reduce to and remember the common conversion table to convert the fraction to 37.5%. The answer

is (D).

2. Use the Bowtie method. First, multiply the denominators: 5 × 12 = 60. When you multiply diagonally, as

in the diagram given in the text, the numerator becomes 35 + 24, or 59. The new fraction is , and the

answer is (D).

3. Simplify the negative exponents by taking the reciprocal of the corresponding positive exponent, which

gives you. Now you have three reciprocals, so flip them over and

calculate: = 59,049. You can also express each number as a power of 3, which gives you

(3^3 )(3^4 )(3^3 ) = 3^10 , which makes (B) correct. 3^10 can also be expressed as. Thus, the correct answer is

(A), (B), and (C).

4. Divide to convert the fractions into decimals. First, = 0.2727. This is really a pattern question: The odd

numbered terms are 2 and the even numbered terms are 7. The 32nd^ digit to the right of the decimal is an

even term, so x = 7. Next, = 0.6363. This time, the odd numbered terms are 6 and the even numbered

terms are 3. The 19th digit on the right side of the decimal place is an odd term, so y = 6. Lastly, xy = 7 × 6

= 42.

5. One-third of 42 is 14, so two-thirds is 28. 28 equals of x translates to 28 = x. Multiply both sides by

the reciprocal, , to eliminate the fraction and isolate the x. The fractions on the right cancel out and on the

left you have (28) = 35, so 35 = x. The answer is (B).

6. To compare two fractions, just cross-multiply and compare the products. 7 × 8 = 56 and 11 × 5 = 55, so

Quantity A is greater.

7. The answer is 0. First, divide to convert into a decimal. The on-screen calculator doesn’t do

exponents, so you may want to factor the denominator: = 0.00003125. Since 0

is even, there are no digits between the decimal point and the first even digit after the decimal point, and the

correct answer is 0. In fact, if you noticed that any decimal starting with a 0 would have the same answer,

you only needed to make sure the denominator was larger than 10.

8. Before you break out the calculator, distribute the (0.000023). This gives you (0.000023)(10^6 ) – (0.000023)(10^4 ). Now move the decimal point 6 places to the right for the first term and 4 places to the right for the second term, which gives you 23.23 − .23 = 23. The correct answer is (D).

PARTS OF A WHOLE (PERCENTS)

As you may have noted from the conversion chart, decimals and percents look an awful lot alike. In fact, all you have to do to convert a decimal to a percent is to move the decimal point two places to the right and add the percent sign: 0.25 becomes 25%, 0.01 becomes 1%, and so forth. This is because they’re both based on multiples of 10. Percents also represent division with a denominator that is always 100.