# Section 5.9: Review for Chapter 5

Studying for a chapter examination is a personal process, one which nobody else can do for you. Simply take the time to review what you have done.

Here are the new terms in Chapter 5.

Abelian group [5.6]
Absolute value [5.3]
Algebra [5.7]
Associative property [5.1]
Canonical form [5.2]
Ciphertext [5.8]
Closed for multiplication [5.1]
Closed set [5.1]
Closure property [5.1]
Commutative group [5.6]
Commutative property [5.1]
Composite number [5.2]
Congruent modulo [5.7]
Counting numbers [5.1]
Cryptography [5.8]
Decoding key [5.8]
Denominator [5.4]
Dense set [5.6]
Discrete mathematics [5.7]
Distributive property [5.1]
Divides [5.2]
Divisibility [5.2]
Division [5.3-5.7]
Division by zero [5.3]
Divisor [5.2]
e [5.5]
Encoding key [5.8]
Encrypt [5.8]
Factor [5.2]
Factor tree [5.2]
Factoring [5.2]
Field [5.6]
Fundamental property of fractions [5.4]
Fundamental theorem of arithmetic [5.2]
g.c.f. [5.2]
Greatest common factor [5.2]
Group [5.6]
Hypotenuse [5.5]
Identity for multiplication [5.6]
Improper fraction [5.4]
Integers [5.3]
Inverse for multiplication [5.6]
Irrational number [5.5]
Laws of square roots [5.5]
l.c.m. [5.2]
Least common denominator [5.4]
Least common multiple [5.2]
Leg of a triangle [5.5]
Modular codes [5.8]
Modulo 5 [5.7]
Multiple [5.2]
Multiplication [5.1; 5.7]
Multiplicative identity [5.6]
Multiplicative inverse [5.6]
Natural numbers [5.1]
Number line [5.6]
Numerator [5.4]
One [5.6]
Opposites [5.6]
Perfect square [5.5]
Pi [5.5]
Positive square root [5.5]
Prime factorization [5.2]
Prime number [5.2]
Proper fraction [5.4]
Pythagorean theorem [5.5]
Rational number [5.4]
Real number line [5.6]
Real numbers [5.6]
Reciprocal [5.6]
Reduced fraction [5.4]
Relatively prime [5.2]
Repeating decimal [5.6]
Rules of divisibility [5.2]
Sieve of Eratosthenes [5.2]
Square number [5.5]
Square root [5.5]
Subtraction [5.1; 5.7]
Terminating decimal [5.6]
Unit distance [5.6]
Whole numbers [5.3, 5.4]
Zero [5.6]
Zero multiplication [5.7]

If you can describe the term, read on to the next one; if you cannot, then look it up in the text (the section number is shown in brackets).

IMPORTANT IDEAS

Can you explain each of these important ideas in your own words?

Properties of numbers:

closure property [5.1];commutative property [5.1]; associative property [5.1];
distributive property [5.1] identity [5.6]; inverse [5.6]; groups [5.6]; Fields [5.6]

Relationships among the following sets of numbers:

natural numbers [5.1]; whole numbers [5.3]; integers [5.3]; rational numbers [5.4]; irrational numbers [5.5]; real numbers [5.6]

Meanings of fundamental operations:

addition, multiplication [5.1]; subtraction [5.1]; division [5.2]; know the difference between a square root and an irrational number [5.5]

Operations among the following sets of numbers:

integers [5.3]; rationals [5.4]; irrationals [5.5]; reals [5.6]
Rules of divisibility [5.2]
Least number of divisors [5.3]
Fundamental theorem of arithmetic [5.3]
Fundamental property of fractions [5.4]
Pythagorean theorem [5.5]
Elementary operations [5.7]

Next, make sure you understand the types of problems in Chapter 5.

TYPES OF PROBLEMS

Demonstrate the definition of multiplication. [5.1]
Determine whether a given set with a given operation is closed. [5.1]
Recognize and distinguish the commutative and associative properties. [5.1]
Apply the distributive property with a variety of operations. [5.1]
Determine whether a given number is prime or not. [5.2]
Tell whether one number divides another number. [5.2]Find the prime factorization and write the answer in canonical form. [5
.2]
Find the least common multiple of a set of numbers. [5.2]
Find the greatest common factor of a set of numbers. [5.2]
Show that there is no largest prime number. [5.2]
Use problem-solving techniques to solve applied problems. [5.2-5.8]
Find the absolute value of a number. [5.3]
Carry out operations with integers. [5.3]
Reduce fractions using the fundamental property of fractions. [5.4]
Carry out operations with fractions. [5.4]
Use the definition of square root to simplify radical expressions. [5.5]
Classify numbers as rational or irrational. [5.5]
Determine into which of the following sets that a given number belongs: natural numbers, integers, rational numbers, irrational numbers, or real numbers. [5.6]
Express a rational number as a decimal. [5.6]
Express a terminating decimal as a fraction. [5.6]
Use the order of operations to simplify real numbers. [5.6]
Find a rational number or an irrational number between each of a given pair of numbers. [5.6]
Recognize and distinguish examples of the closure, associative, commutative, identity, and inverse properties. [5.6]
Know and be able to describe each of the field properties. [5.6]
Carry out operations in modular arithmetic. [5.7]
Solve modular equation. [5.7]
Decide if a given set and operation forms a group. [5.7]
Decide if a given set and two operations forms a field. [5.7]
Encode and decode simple phrases. [5.7]
Break a simple code. [5.7]

Once again, see if you can verbalize (to yourself) how to do each of the listed types of problems. Work all of Chapter 5 Review Questions (whether they are assigned or not).

Work through all of the problems before looking at the answers, and then correct each of the problems. The entire solution is shown in the answer section at the back of the text. If you worked the problem correctly, move on to the next problem, but if you did not work it correctly (or you did not know what to do), look back in the chapter to study the  procedure, or ask your instructor. Finally, go back over the homework problems you have been assigned. If you worked a problem correctly, move on the next problem, but if you missed it on your homework, then you should look back in the text or talk to your instructor about how to work the problem. If you follow these steps, you should be
successful with your review of this chapter.

We give all of the answers to the Chapter Review questions (not just the odd-numbered questions), so be sure to check your work with the answers as you prepare for an examination.