Transcript Math 191: Mathematics and Geometry for Designers

Math 191: Mathematics and
Geometry for
Designers
Lecture Notes Based on The Book
Mathematics in Our World
by
Allan G. Bluman
Chapter 1
The Real Number System
1.1 The Natural Numbers
1.1.1 Prime and Composite Numbers
1.1.2 Prime Factorizations
Every 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 Integers
1.2.1 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 with
the 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 Integers
Although, 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 values
of 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 of
the number being subtracted and follow the rules of addition.
1.2.3 Multiplication and Division of Integers
Multiplication is a shortcut for addition. For instance, 3×2 means to add 2 three
times (or to add 3 two times). Since integers also involve negative numbers, we
follow 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 multiplication
of 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 grouping
symbols 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 Numbers
1.3.1 Definition of Rational Numbers
1.5 The Real Numbers
1.5.1 Definition of Real Numbers
1.6 Exponents and Scientific Notations
1.6.1 Properties of Exponents