Bob Miller - Bob Millers Math for the Accuplacer

I have many people to thank.

I thank my wife, Marlene, who makes life worth living, who is truly the wind under my wings.

I thank the rest of my family: children Sheryl and Eric and their spouses Glenn and Wanda (who are also like my children); grandchildren Kira, Evan, Sean, Sarah, Ethan, and Noah; my brother Jerry; and my parents, Cele and Lee; and my in-law parents, Edith and Siebeth.

I thank Larry Kling and Michael Reynolds and Mel Friedman for making this book possible.

I thank Martin Levine for making my whole writing career possible.

I have been negligent in thanking my great math teachers of the past. I thank Mr. Douglas Heagle, Mr. Alexander Lasaka, Mr. Joseph Joerg, and Ms. Arloeen Griswold, the best math teacher I ever had, of George W. Hewlett High School; Ms. Helen Bowker of Woodmere Junior High; and Professor Pinchus Mendelssohn and Professor George Bachman of Polytechnic University. The death of Professor Bachman was an extraordinary loss to our country, which produces too few advanced degrees in math. Every year, two or three of Professor Bachmans students would receive a Ph.D. in math, and even more would receive their M.S. in math. In addition, he wrote four books and numerous papers on subjects that had never been written about or had been written so poorly that nobody could understand the material. His teachings and writings were clear and memorable.

As usual, the last three thanks go to three terrific people: a great friend, Gary Pitkofsky; another terrific friend and fellow lecturer, David Schwinger; and my cousin, Keith Robin Ellis, the sharer of our dreams.

I n addition to our author, we would like to thank Larry B. Kling, Vice President, Editorial, for his overall direction; Pam Weston, Vice President, Publishing, for setting the quality standards for production integrity and managing the publication to completion; Michael Reynolds, Managing Editor, for project management; Mel Friedman, Lead Mathematics Editor, for editorial contributions; Rachel DiMatteo, Graphic Designer, for designing this book; Christine Saul, Senior Graphic Artist, for designing our cover; and Datastream Content Solutions, for typesetting this edition. Back-cover photo of author by Eric L. Miller.

OTHER BOOKS

Bob Millers Math for the ACT

Bob Millers Math for the GMAT

Bob Millers Math for the GRE

Bob Millers Math for the TABE

Bob Millers Basic Math and Pre-Algebra for the Clueless, Second Edition

Bob Millers Algebra for the Clueless, Second Edition

Bob Millers Geometry for the Clueless, Second Edition

Bob Millers Math SAT for the Clueless, Second Edition

Bob Millers Pre-Calc with Trig for the Clueless, Third Edition

Bob Millers High School Calc for the Clueless

Bob Millers Calc 1 for the Clueless, Second Edition

Bob Millers Calc 2 for the Clueless, Second Edition

Bob Millers Calc 3 for the Clueless

NOTES

CHAPTER 1:

The Beginnings

You have a 60-inch ribbon. If you cut it so that each piece is 6 inches, how many cuts do you make?

Our great adventure begins with our basic terms. It is very important to understand what a question asks as well as how to answer it. The word numbers has many meanings, as we start to see here.

NUMBERS

Whole numbers: 0, 1, 2, 3, 4,...

Integers: 0, 1, 2, 3, 4,... , where 3 stands for both +3 and 3.

Positive integers are integers that are greater than 0. In symbols, x > 0, x an integer.

Negative integers are integers that are less than 0. In symbols, x < 0, x an integer.

Even integers: 0, 2, 4, 6,...

Odd integers: 1, 3, 5, 7,...

Inequalities

For any numbers represented by a , b , c , or d on the number line:

We say c > d ( c is greater than d ) if c is to the right of d on the number line.

We say d < c ( d is less than c ) if d is to the left of c on the number line.

c > d is equivalent to d < c .

a < b means a < b or a = b ; likewise, a > b means a > b or a = b .

Example 1:

4 < 7 is true because 4 < 7; 9 < 9 is true because 9 = 9; but 7 < 2 is false because 7 > 2.

Example 2:

Find all integers between 4 and 5.

Solution:

{ 3, 2, 1, 0, 1, 2, 3, 4}.

Notice that the word between does not include the endpoints.

Example 3:

Graph all the multiples of five between 20 and 40 inclusive.

Solution:

Notice that inclusive means to include the endpoints.

Odd and Even Numbers

Here are some facts about odd and even integers that you should know.

• The sum of two even integers is even.
• The sum of two odd integers is even.
• The sum of an even integer and an odd integer is odd.
• The product of two even integers is even.
• The product of two odd integers is odd.
• The product of an even integer and an odd integer is even.
• If n is even, n 2 is even. If n 2 is even and n is an integer, then n is even.
• If n is odd, n 2 is odd. If n 2 is odd and n is an integer, then n is odd.

OPERATIONS ON NUMBERS

Product is the answer in multiplication, quotient is the answer in division, sum is the answer in addition, and difference is the answer in subtraction.

Because 3 4 = 12, 3 and 4 are said to be factors or divisors of 12, and 12 is both a multiple of 3 and a multiple of 4.

A prime is a positive integer with exactly two distinct factors, itself and 1. The number 1 is not a prime because only 1 1 = 1. It might be a good idea to memorize the first eight primes:

2, 3, 5, 7, 11, 13, 17, and 19

The number 4 has more than two factors:1, 2, and 4. Numbers with more than two factors are called composites. The number 28 is a perfect number because if we add the factors less than 28, they add to 28.

Example 4:

Write all the factors of 28.

Solution:

1, 2, 4, 7, 14, and 28.

Example 5:

Write 28 as the product of prime factors.

Solution:

28 = 2 2 7.

Example 6:

Find all the primes between 70 and 80.

Solution:

71, 73, 79. How do we find this easily? First, because 2 is the only even prime, we have to check only the odd numbers. Next, we have to know the divisibility rules:

• A number is divisible by 2 if it ends in an even number. We dont need this here because then it cant be prime.