Math 191: Mathematics and Geometry forDesigners

Lecture Notes Based on The BookMathematics in Our World

byAllan G. Bluman

Chapter 1The Real Number System

1.1 The Natural Numbers1.1.1 Prime and Composite Numbers

1.1.2 Prime FactorizationsEvery composite number can be expressed as a product of prime numbers in only one way. For example, 24 can be written in primes as 2 ×2×2×3. There is no other way to write 24 as a product of primes. This product is called a prime factorization of 24.

Theorem 6 (Fundamental Theorem of Arithmetic) Every composite number can be expressed as a product of primes in only one way regardless of order of the prime factors.

Example 7 Let us try to find a prime factorization of 100. First, divide 100 by 2 and then divide the answer by 2. Continue dividing the answer until you cannot find an answer that is not divisible by 2, then try to divide by 3, then 5, etc. Following the procedure introduced, it is not difficult to see that 100 = 2 × 2 × 5 × 5 = 22 × 52.

1.2 The Integers1.2.1 Definition of Integers

• Definition 13 The set of whole numbers, denoted by W, consists of

0, 1, 2, 3, . . . .When the numbers −1, −2, −3, . . . are included withthe set of whole numbers, a new set is formed.• Definition 14 The set of integers, denoted by Z,

consists of. . . , −3, −2, −1, 0, 1, 2, 3, . . . .

1.2.2 Addition and Subtraction of IntegersAlthough, one can use the number line to add/subtract integers, the following rules are more handful. Rule 1. To add two integers with the same signs, add the absolute valuesof the numbers and give the answer the common sign.Rule 2. To add two integers with different signs, subtract the number with the smaller absolute value from the number with the larger absolute value and give the answer the sign of the number with the larger absolute value.

Remark 18 When performing subtraction using integers, change the sign ofthe number being subtracted and follow the rules of addition.

1.2.3 Multiplication and Division of IntegersMultiplication is a shortcut for addition. For instance, 3×2 means to add 2 threetimes (or to add 3 two times). Since integers also involve negative numbers, wefollow the following simple rules when we multiply to integers.Rule 1. The product of two numbers with like signs is positive.Rule 2. The product of two numbers with unlike signs is negative.

The rules for dividing integers follow the same pattern as the rules for multiplicationof integers.Rule 1. The quotient of two numbers with like signs is positive.Rule 2. The quotient of two numbers with unlike signs is negative.

Grammar rules and punctuation symbols clarify the meaning of sentences. Mathematics also has rules and symbols that clarify the meaning of expression. These rules are called the order of operations. We adapt the following simple steps for the order of operations.

Step 1. Perform all calculations inside grouping symbols first. The groupingsymbols used in mathematics are parentheses ( ), brackets [ ] and braces { }.Step 2. Evaluate all exponents.Step 3. Perform all multiplication and division in order from left to right.Step 4. Perform all addition and subtraction in order from left to right.

1.3 The Rational Numbers1.3.1 Definition of Rational Numbers

1.5 The Real Numbers1.5.1 Definition of Real Numbers

1.6 Exponents and Scientific Notations1.6.1 Properties of Exponents