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Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. Chapter 1 Number Theory and the Real Number System An introduction to number theory Prime numbers Integers, rational numbers, irrational numbers, and real numbers Properties of real numbers Rules of exponents and scientific notation WHAT YOU WILL LEARN

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Slide 5 - 2 Copyright © 2009 Pearson Education, Inc. 1.1 Number Theory The study of numbers and their properties. The numbers we use to count are called natural numbers, or counting numbers. N = { 1, 2, 3, 4, } The natural numbers that are multiplied together to equal another natural number are called factors of the product. Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

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Slide 5 - 3 Copyright © 2009 Pearson Education, Inc. Divisors If a and b are natural numbers and the quotient of b divided by a has a remainder of 0, then we say that a is a divisor of b or a divides b. Prime and Composite Numbers A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1. A composite number is a natural number that is divisible by a number other than itself and 1. The number 1 is neither prime nor composite, it is called a unit.

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Slide 5 - 4 Copyright © 2009 Pearson Education, Inc. The Fundamental Theorem of Arithmetic Every composite number can be expressed as a unique product of prime numbers. This unique product is referred to as the prime factorization of the number. There are two ways to do this as we show in the next two slides.

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Slide 5 - 5 Copyright © 2009 Pearson Education, Inc. Example of branching method for the number 3190 Therefore, the prime factorization of 3190 =

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Slide 5 - 6 Copyright © 2009 Pearson Education, Inc. Example of division method Write the prime factorization of 663. The prime factorization of 663 is

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Slide 5 - 7 Copyright © 2009 Pearson Education, Inc. Finding the LCM of Two or More Numbers Determine the prime factorization of each number. List each prime factor with the greatest exponent that appears in any of the prime factorizations. Determine the product of the factors found in step 2. Example (LCM) Find the LCM of 63 and 105. So, the LCM is

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Slide 5 - 8 Copyright © 2009 Pearson Education, Inc. Example of LCM Find the LCM of 48 and 54. Prime factorizations of each: 48 = 54 = LCM =

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Slide 5 - 9 Copyright © 2009 Pearson Education, Inc. 1.2 Integers The set of integers consists of 0, the natural numbers, and the negative natural numbers. Integers = {…–4, –3, –2, –1, 0, 1, 2, 3 4,…} On a number line, the positive numbers extend to the right from zero; the negative numbers extend to the left from zero.

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Slide 5 - 10 Copyright © 2009 Pearson Education, Inc. Addition of Integers Subtraction of Integers Multiplication of Integers Division of Integers Will all be done using your calculator. Do not confuse the (-) sign with the subtraction sign when doing these calculations!

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Slide 5 - 11 Copyright © 2009 Pearson Education, Inc. Reducing Fractions In order to reduce a fraction to its lowest terms, we divide both the numerator and denominator by the greatest common divisor. Example: Reduce to its lowest terms. Solution:

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Slide 5 - 12 Copyright © 2009 Pearson Education, Inc. Improper Fractions Rational numbers greater than 1 or less than – 1 that are not integers may be written as mixed numbers, or as improper fractions. An improper fraction is a fraction whose numerator is greater than its denominator. An example of an improper fraction is. To write this as a mixed number we write it as 2

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Slide 5 - 13 Copyright © 2009 Pearson Education, Inc. We can do the reverse process: Convert to an improper fraction. An Example of going the other way: Convert to a mixed number.

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Slide 5 - 14 Copyright © 2009 Pearson Education, Inc. Multiplication of Fractions Division of Fractions

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Slide 5 - 15 Copyright © 2009 Pearson Education, Inc. Example: Multiplying Fractions Evaluate the following. a) b)

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Slide 5 - 16 Copyright © 2009 Pearson Education, Inc. Example: Dividing Fractions Evaluate the following. a) b)

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Slide 5 - 17 Copyright © 2009 Pearson Education, Inc. Example: Add or Subtract Fractions with alike denominators Add: Subtract:

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Slide 5 - 18 Copyright © 2009 Pearson Education, Inc. Example with unalike denominators: Evaluate: Solution: first find the LCD:

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Slide 5 - 19 Copyright © 2009 Pearson Education, Inc. Irrational Numbers An irrational number is a real number whose decimal representation is a nonterminating, nonrepeating decimal number. Examples of irrational numbers: 5.12639573… 6.1011011101111… 0.523225222… Please note that these are different from a repeating decimal as shown in the next slide.

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Slide 5 - 20 Copyright © 2009 Pearson Education, Inc. Terminating or Repeating Decimal Numbers Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number. Examples of terminating decimal numbers are 0.7, 2.85, 0.000045 Examples of repeating decimal numbers 0.44444… which may be written And 0.23232323… which can be written

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Slide 5 - 21 Copyright © 2009 Pearson Education, Inc. Perfect Square Any number that is the square of a natural number is said to be a perfect square. The numbers 1, 4, 9, 16, 25, 36, and 49 are the first few perfect squares.

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Slide 5 - 22 Copyright © 2009 Pearson Education, Inc. Radicals are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand. The principal (or positive) square root of a number n, is the positive number that when multiplied by itself, gives n. For example, since and since

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Slide 5 - 23 Copyright © 2009 Pearson Education, Inc. Product Rule for Radicals Simplify: a) b)

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Slide 5 - 24 Copyright © 2009 Pearson Education, Inc. Example: Adding or Subtracting Irrational Numbers Simplify:

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Slide 5 - 25 Copyright © 2009 Pearson Education, Inc. Multiplication of Radicals

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Slide 5 - 26 Copyright © 2009 Pearson Education, Inc. Example: Division Divide: Solution: Divide: Solution:

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Slide 5 - 27 Copyright © 2009 Pearson Education, Inc. A denominator is rationalized when it contains no radical expressions. Rationalize the denominator of Solution:

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Slide 5 - 28 Copyright © 2009 Pearson Education, Inc. Commutative Property Addition a + b = b + a for any real numbers a and b. 8 + 12 = 12 + 8 is a true statement. 5 9 = 9 5 is a true statement. Note: The commutative property does not hold true for subtraction or division. Multiplication a b = b a for any real numbers a and b.

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Slide 5 - 29 Copyright © 2009 Pearson Education, Inc. Associative Property Addition (a + b) + c = a + (b + c), for any real numbers a, b, and c. Multiplication (a b) c = a (b c), for any real numbers a, b, and c.

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Slide 5 - 30 Copyright © 2009 Pearson Education, Inc. Example (3 + 5) + 6 = 3 + (5 + 6) is true. (4 6) 2 = 4 (6 2) is true. Note: The associative property does not hold true for subtraction or division.

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Slide 5 - 31 Copyright © 2009 Pearson Education, Inc. Distributive Property Distributive property of multiplication over addition a (b + c) = a b + a c for any real numbers a, b, and c. Example: 6 (r + 12) =

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Slide 5 - 32 Copyright © 2009 Pearson Education, Inc. 1.6 Exponents and Scientific Notation When a number is written with an exponent, there are two parts to the expression: base exponent The exponent tells how many times the base should be multiplied together.

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Slide 5 - 33 Copyright © 2009 Pearson Education, Inc. Rules of exponents Product rule: Simplify: 3 4 3 9 3 4 3 9 = 3 4 + 9 = 3 13 Simplify: 6 4 6 5 6 4 6 5 = 6 4 + 5 = 6 9 Quotient Rule: Simplify: Simplify Power Rule: Simplify: (3 2 ) 3 Simplify: (2 3 ) 5 Zero exponent rule: Simplify: Negative Exponent Rule: Simplify :

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Slide 5 - 34 Copyright © 2009 Pearson Education, Inc. Scientific Notation Many scientific problems deal with very large or very small numbers. 93,000,000,000,000 is a very large number. 0.000000000482 is a very small number. Scientific notation is a shorthand method used to write these numbers. 9.3 10 13 and 4.82 10 –10 are two examples of numbers in scientific notation. Write each number in scientific notation. a)1,265,000,000. b)0.000000000432

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Slide 5 - 35 Copyright © 2009 Pearson Education, Inc. Example going the other way: Write each number in decimal notation. a)4.67 10 5 b)1.45 10 –7

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