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Math 150 - Chapter 1

Math 150 - Chapter 1. Number Theory and the Real Number System Theodore Vassiliadis. WHAT YOU WILL LEARN. An introduction to number theory • Prime numbers • Integers, rational numbers, irrational numbers, and real numbers • Properties of real numbers

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Math 150 - Chapter 1

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  1. Math 150 - Chapter 1 Number Theory and the Real Number System Theodore Vassiliadis

  2. WHAT YOU WILL LEARN 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

  3. Number Theory • The study of numbers and their properties. • The numbers we use to count are called natural numbers, , or counting numbers.

  4. Factors • 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.

  5. 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.

  6. 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.

  7. Rules of Divisibility Divisible by Test Example 2 The number is even. 846 3 The sum of the digits of the number is divisible by 3. 846 since 8 + 4 + 6 = 18 4 The number formed by the last two digits of the number is divisible by 4. 844 since 44  4 5 The number ends in 0 or 5. 285

  8. Divisible by Test Example 6 The number is divisible by both 2 and 3. 846 8 The number formed by the last three digits of the number is divisible by 8. 3848 since 848  8 9 The sum of the digits of the number is divisible by 9. 846 since 8 + 4 + 6 = 18 10 The number ends in 0. 730 Divisibility Rules, continued

  9. 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.

  10. Division Method 1. Divide the given number by the smallest prime number by which it is divisible. 2. Place the quotient under the given number. 3. Divide the quotient by the smallest prime number by which it is divisible and again record the quotient. 4. Repeat this process until the quotient is a prime number.

  11. 3 663 13 221 17 Example of division method • Write the prime factorization of 663. • The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is 3 •13 •17

  12. 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.

  13. Example (LCM) • Find the LCM of 63 and 105. 63 = 32 • 7 105 = 3 • 5 • 7 • Greatest exponent of each factor: 32, 5 and 7 • So, the LCM is 32 • 5 • 7 = 315.

  14. Whole Numbers • The set of whole numbers contains the set of natural numbers and the number 0. • Whole numbers = {0,1,2,3,4,…}

  15. 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.

  16. Writing an Inequality • Insert either > or < in the box between the paired numbers to make the statement correct. a) 3 1 b) 9 7 3 < 1 9 < 7 c) 0 4 d) 6 8 0 > 4 6 < 8

  17. 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:

  18. Fractions • Fractions are numbers such as: • The numerator is the number above the fraction line. • The denominator is the number below the fraction line.

  19. 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:

  20. 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

  21. Multiplication of Fractions • Division of Fractions

  22. Example: Multiplying Fractions • Evaluate the following. a) b)

  23. Example: Dividing Fractions • Evaluate the following. a) b)

  24. Addition and Subtraction of Fractions

  25. Example: Add or Subtract Fractions Add: Subtract:

  26. Fundamental Law of Rational Numbers • If a, b, and c are integers, with b 0, c  0, then

  27. Example: • Evaluate: • Solution:

  28. Irrational Numbers • An irrational number is a real number whose decimal representation is a nonterminating, nonrepeating decimal number. • Examples of irrational numbers:

  29. Radicals • are all irrational numbers. The symbol is called the radicalsign. The number or expression inside the radical sign is called the radicand.

  30. 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.

  31. Real Numbers • The set of real numbers is formed by the union of the rational and irrational numbers. • The symbol for the set of real numbers is

  32. Real numbers Rational numbers Integers Irrational numbers Whole numbers Natural numbers Relationships Among Sets

  33. Properties of the Real Number System • Closure If an operation is performed on any two elements of a set and the result is an element of the set, we say that the set is closed under that given operation.

  34. Commutative Property • Addition a + b = b + a for any real numbers a and b. • Multiplication a • b = b •a for any real numbers a and b.

  35. Example • 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.

  36. 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.

  37. 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.

  38. 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) = 6 • r + 6 • 12 = 6r + 72

  39. Exponents • When a number is written with an exponent, there are two parts to the expression: baseexponent • The exponent tells how many times the base should be multiplied together.

  40. 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.

  41. Scientific Notation continued • Scientific notation is a shorthand method used to write these numbers. • 9.3  1013 and 4.82  10–10 are two examples of numbers in scientific notation.

  42. To Write a Number in Scientific Notation 1. Move the decimal point in the original number to the right or left until you obtain a number greater than or equal to 1 and less than 10. 2. Count the number of places you have moved the decimal point to obtain the number in step 1. If the decimal point was moved to the left, the count is to be considered positive. If the decimal point was moved to the right, the count is to be considered negative. 3. Multiply the number obtained in step 1 by 10 raised to the count found in step 2. (The count found in step 2 is the exponent on the base 10.)

  43. Example • Write each number in scientific notation. a) 1,265,000,000. 1.265  109 b) 0.000000000432 4.32  1010

  44. To Change a Number in Scientific Notation to Decimal Notation 1. Observe the exponent on the 10. 2. a) If the exponent is positive, move the decimal point in the number to the right the same number of places as the exponent. Adding zeros to the number might be necessary. b) If the exponent is negative, move the decimal point in the number to the left the same number of places as the exponent. Adding zeros might be necessary.

  45. Example • Write each number in decimal notation. a)4.67  105 467,000 b) 1.45  10–7 0.000000145

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