Slide Copyright © 2009 Pearson Education, Inc. Unit 1 Number Theory MM-150 SURVEY OF MATHEMATICS – Jody Harris

Slide Copyright © 2009 Pearson Education, Inc. Number Theory Number theory is the study of natural numbers and their properties. The numbers we use to count are called natural numbers,, or counting numbers.

Slide Copyright © 2009 Pearson Education, Inc. Factors and Divisors The natural number a is a factor of b if there is another natural number k such that a  k = b Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Factors of a natural number are also called divisors because if we divide a natural number by one if its factors the remainder is 0.

Slide Copyright © 2009 Pearson Education, Inc. Prime and Composite Numbers A prime number is a natural number that has exactly two factors (or divisors), itself and 1.

Slide Copyright © 2009 Pearson Education, Inc. Prime and Composite Numbers A prime number is a natural number 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; in other words it has more than 2 divisors.

Slide Copyright © 2009 Pearson Education, Inc. Prime and Composite Numbers A prime number is a natural number 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; in other words it has more than 2 divisors. The number 1 is neither prime nor composite, it is called a unit.

Slide Copyright © 2009 Pearson Education, Inc. Prime and Composite Numbers A prime number is a natural number 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; in other words it has more than 2 divisors. The number 1 is neither prime nor composite, it is called a unit. Example: Which numbers are prime and which are composite? 2, 3, 9, 13, 22

Slide Copyright © 2009 Pearson Education, Inc. Some Rules for Divisibility 8 2: Any even number, that is, any number that ends in 0, 2, 4, 6, or 8. 3: If the sum of its digits is divisible by 3, for example = 15, which is divisible by 3, so 2346 is. 5: If it ends in 0 or 5. 6: If it is divisible by both 2 AND 3, so 2346 is. 9: If the sum of its digits is divisible by 9, for example = 18, which is divisible by 9, so 2547 is. 10: If it ends in 0.

Slide 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.

Slide 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.

Slide 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.

Slide 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.

Slide 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.

Slide Copyright © 2009 Pearson Education, Inc. Prime Factorization using a Factor Tree 2  3 2

Slide Copyright © 2009 Pearson Education, Inc. Prime Factorization using a Factor Tree 2   3

Slide Copyright © 2009 Pearson Education, Inc. Greatest Common Divisor The greatest common divisor (GCD) of a set of natural numbers is the largest natural number that divides (without remainder) every number in that set.

Slide Copyright © 2009 Pearson Education, Inc. Finding the GCD of Two or More Numbers Determine the prime factorization of each number. List each common prime factor with smallest exponent that appears in each of the prime factorizations. The product of the factors found in the previous step are the GCD.

Slide Copyright © 2009 Pearson Education, Inc. Example (GCD) Find the GCD of 63 and =

Slide Copyright © 2009 Pearson Education, Inc. Example (GCD) Find the GCD of 63 and = =

Slide Copyright © 2009 Pearson Education, Inc. Example (GCD) Find the GCD of 63 and = = 3 5 7

Slide Copyright © 2009 Pearson Education, Inc. Example (GCD) Find the GCD of 63 and = 3 2  = 3  5  7 Smallest exponent of each common factor: 3 and 7

Slide Copyright © 2009 Pearson Education, Inc. Example (GCD) Find the GCD of 63 and = = Smallest exponent of each common factor: 3 and 7 So, the GCD is 3 7 = 21.

Slide Copyright © 2009 Pearson Education, Inc. Least Common Multiple The least common multiple (LCM) of a set of natural numbers is the smallest natural number that is divisible (without remainder) by each element of the set.

Slide 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. The product of the factors found in step 2 is the LCM.

Slide Copyright © 2009 Pearson Education, Inc. Example (LCM) Find the LCM of 63 and = = 3 5 7

Slide Copyright © 2009 Pearson Education, Inc. Example (LCM) Find the LCM of 63 and = = Greatest exponent of each factor: 3 2, 5 and 7

Slide Copyright © 2009 Pearson Education, Inc. Example (LCM) Find the LCM of 63 and = = Greatest exponent of each factor: 3 2, 5 and 7 So, the LCM is = 315.

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

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

Slide Copyright © 2009 Pearson Education, Inc. Example of GCD and LCM Find the GCD and LCM of 48 and 54. Prime factorizations of each: 48 = = = = 2 3 3

Slide Copyright © 2009 Pearson Education, Inc. Example of GCD and LCM Find the GCD and LCM of 48 and 54. Prime factorizations of each: 48 = = = = GCD = 2 3 = 6

Slide Copyright © 2009 Pearson Education, Inc. Example of GCD and LCM Find the GCD and LCM of 48 and 54. Prime factorizations of each: 48 = = = = GCD = 2 3 = 6 LCM = = 432

Slide Copyright © 2009 Pearson Education, Inc. Whole Numbers The set of whole numbers contains the set of natural numbers and the number 0. Whole numbers = {0,1,2,3,4,…}

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. The Rational Numbers The set of rational numbers, denoted by Q, is the set of all numbers of the form p/q, where p and q are integers and q  0. The following are examples of rational numbers:

Slide 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, Examples of repeating decimal numbers … which may be written

Slide 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.

Slide 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:

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

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

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

Slide Copyright © 2009 Pearson Education, Inc. Addition and Subtraction of Fractions

Slide Copyright © 2009 Pearson Education, Inc. Fundamental Law of Rational Numbers If a, b, and c are integers, with b  0, c  0, then

Slide Copyright © 2009 Pearson Education, Inc. Example: Evaluate: Solution:

Slide 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:

Slide 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.

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

Slide Copyright © 2009 Pearson Education, Inc. Addition and Subtraction of Irrational Numbers To add or subtract two or more square roots with the same radicand, add or subtract their coefficients. The answer is the sum or difference of the coefficients multiplied by the common radical.

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

Slide Copyright © 2009 Pearson Education, Inc. Multiplication of Irrational Numbers Simplify:

Slide Copyright © 2009 Pearson Education, Inc. Quotient Rule for Radicals

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

Slide Copyright © 2009 Pearson Education, Inc. Rationalizing the Denominator A denominator is rationalized when it contains no radical expressions. To rationalize the denominator, multiply BOTH the numerator and the denominator by a number that will result in the radicand in the denominator becoming a perfect square. Then simplify the result.

Slide Copyright © 2009 Pearson Education, Inc. Example: Rationalize Rationalize the denominator of Solution: