1. Unit 1 MM 150: Number Theory and the Real Number System Prof. Carolyn Dupee July 3, 2012

2. IMPORTANT NUMBER TERMS Number Theory- the study of numbers and their properties using counting, or real numbers. Prime Numbers- a natural number greater than 1 that has exactly two different factors (or divisors), the number itself and 1. Ex. 2, 3, 5, 7, 11, 13, 17, 19, 23 Composite Numbers- a natural number that is divisible by a number other than one and itself. Ex. 4, 8, 12,15, 16, 20, The number 1 is a special case. It is NOT considered to be a prime number or composite number. 2

3. RULES OF DIVISIBILITY Is the number divisible by 2 evenly? Yes. Consider other #s. -Ex. 1216 ÷ 2 = 608 Does the number end in a 0? If so, it’s divisible by 10. -Ex. 1200 ÷ 10 =120 Can the last 2 numbers be divided evenly by 4? 216 -Ex. 1216 (16 ÷ 2 =8) Yes, it’s divisible by 2. Can you add the digits up in the number so that the number is evenly divisible by 3? If so, the # is divisible by 3. -Ex. 918= 9 + 1 + 8 = 18 ÷ 3 = 6 Yes, it’s divisible by 3. 3

4. RULES OF DIVISIBILITY If a number is divisible by both 2 and 3, then it is divisible by 6. Can the last 2 numbers be divided evenly by 4? -Ex. 1216 (16 ÷ 4 = 4) Yes, it’s divisible by 4 Can the last 3 numbers be divided evenly by 8? -Ex. 216 ÷ 8 = 27 Yes, it’s divisible by 8 Does it end in 0 or 5? -Ex. 2435 Yes, it is divisible by 5. Can the sum of the digits be added up to be divisible evenly by 9? -Ex. 918 (9 + 1+ 8 = 18 ÷ 9 = 2) Yes, it is divisible by 9. 4

5. PRIME FACTORIZATION Breaking down composite numbers into their prime factors using branching (factor trees) or division. 10001000 ÷ 2= 500 ÷ 2 = 250 ÷ 2 = 125 ÷ 5 = 25 ÷ 5 10100 2 5 10 10 2 5 2 5 Use the smallest factors to determine the factors and combine them if you can. 2 X 5 X 2 X 5 X 2 X 5 (a dot, a X, or * can be used for multiplication) Because I have three of the number 2, I will combine it to a power. 2 3 = 8 Because I have three 5’s, I will combine it to a power 5 3 = 125 Then, I will show them multiplied together. 2 3 5 3 5

6. GREATEST COMMON DIVISOR/FACTOR (GCD/F) Abbreviated GCD or GCF The largest natural number that divides every number in the set. -Find the common smallest prime #s (p. 7 from your book Examples 4 and 5) -Multiply common numbers once (I’ve put the common numbers in red) 12 18 6 2 3 6 2 3 2 3Both the 12 and 18 have the #’s 2 and 3 in common so multiply them together 2 3= 6 The GCD or GCF is 6. 6

7. LEAST COMMON MULTIPLE (LCM) The smallest natural number that is divisible by each element of the set. It is a number that is larger than both of the 2 numbers you are considering (UNLESS- it is one of the numbers shown) 12 18 6 2 3 6 2 3 See the common prime #s like before. You will use these numbers to help you determine the LCM.. Multiply them together just once. You also must consider the smallest prime factors from each number that the number has unique to itself. For the number 12 (2) and for the number 18 (3). Multiply all red and blue numbers together. 2 3 2 3 = 36 is your LCM 7

8. WRITING INEQUALITILES > means greater than and < means less than The large open part or “mouth” always points towards the larger number. I always pretend that the mouth is for an alligator. -Ex. 2 > -1 -Ex. 4 < 10 8

9. INTEGERS Negative numbers, 0, and positive numbers Adding Integers 4 + (+4) = 8-4 + 10 = +6 4 + (-4) = 0-4 + -2 = -6 Subtracting Integers 4 – 4 = 0-4 – 4 = -8 4 –(-4) = 0-4 + 4 = 0 Even # of negatives in a row, they cancel out. Two negatives equal a positive. 9

10. INTEGERS CONTINUED Multiplication of Integers 2 (3) = 6 () are another multiplication notation 2 (-3) = -6 (odd number of negatives, keep the negative) -2 (3) = -6 -2 (-3) = + 6 (even number of positives, they cancel out, you’re really saying the opposite of the negative). 10

11. INTEGERS CONTINUED (DIVIDING INTEGERS) 6 = +2-6 = -26 = -2-6 = +2 3 3 -3 -3 Again, odd number of negatives, keep the sign. Even number of negatives, cancel out and two negatives give you a positive. 11

12. RATION NUMBERS One example is a fraction. Numerator = 28 ÷ 2 = 14 Denominator 34 ÷ 2 = 17 Reducing Fractions to Lowest Terms: whatever you do to the numerator, you must do to the denominator. Use divisibility rules and GCD. In this case, you can reduce both once by 2 and you end up with 14/17. 12

13. MIXED NUMBER TO IMPROPER FRACTION Mixed numbers are whole numbers with fractions. Multiply the larger number by the bottom number (denominator). Add the numerator (top number). Put the new number over the original denominator. When converting between mixed number and improper fractions, the denominator will never change. Pg. 24 Ex. 2a) 1 2 (1X 3= 3 + 2 = 5) 5 3 3 13

14. IMPROPER FRACTION TO MIXED NUMBER Divide the numerator by the denominator. Put the remainder over the original denominator. Again, the denominator won’t chance when we do these conversions. Put your answer from dividing with your fraction. 20 = 6 with 2 left over 6 2 3 3 14

15. TERMINATING AND REPEATING DECIMALS Pg. 26 & 27 Ex. 5 & 6 5A) ¾ = 0.75 This is terminating because it can be divided evenly after a few places. 5B) – 7/20 = -0.35 5C) 13/8 = 1.625 6a. 2/3 = 2÷ 3 = 0.6666666 = O.6 with a line over the first 6 because that is the number that’s repeated. 6b. 14/99= 14 ÷ 99= 0.14141414, in this case your line is over the 14. 15

16. CONVERTING DECIMALS TO FRACTIONS Read the place value of the fraction where it ends. See p. 27 for place values. Put it over the proper units. Reduce if possible. Ex. A). 0.7 = 7 tenths = 7 10 Ex. B) 0.35 = 35 hundredths = 35 ÷ 5 = 7 100 ÷ 5 = 20 16

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18. CONVERTING DECIMALS TO FRACTIONS Ex. 7c). 0.016 = 16 thousandths 16 ÷ 8 =2 1000 ÷ 8 125 18

19. MULTIPLYING FRACTIONS Make mixed numbers into improper fractions. Multiply straight across and reduce if possible. (Possibly use cross cancelling which needs to be done diagonally or with a numerator and a denominator from the same fraction). A) 1 X 2 = 2 ÷ 2 = 1 OR 4 3 12 ÷ 2 = 6 1 X 2 1 = 1 4 2 3 6 19

20. MULTIPLYING MIXED # FRACTIONS 2 2 X 1 = 8 X 1 = 8 ÷ 4 = 2 3 4 3 4 12 ÷ 4 3 2 2 X 1 = 2 8 X 1 = 2 3 4 3 4 1 3 20

21. DIVIDING FRACTIONS Invert the second fraction and multiply! 1 ÷ 2 = 1 X 3 = 3 4 3 4 2 8 2 1 ÷ 3 = 9 X 4 = 9 = 3 4 4 4 3 3 21

22. ADDING AND SUBTRACTING FRACTIONS Find a common denominator. Make fractions equivalent (same denominator). Add or subtract numbers keeping numerators over denominators. Reduce if possible. 22

23. ADDING AND SUBTRACTING FRACTIONS 1 4 = 42 1 2= 2 3 2 13= 3+12= 15 3 4 12 42= 8 43= 12+12 12 -1 3 - 3+3 7= 7 - 2 14=4 -4 4 3 12 8 8 34 12 12 ----- ------------------------- 15 9 = 5 + 1 +1 = 6 111 12 8 8 812 23

24. IRRATIONAL NUMBERS A real number whose decimal representation is nonterminating, nonrepeating decimal. Square roots and the number pi. The table on the next page is helpful to remember the realtionship to make your life easier. You can still use your calculator to determine these values, but you will be using these numbers a lot and it’s helpful to know them, or to be able to refer to them. 24

25. IRRATIONAL NUMBERS Square roots √Perfect Square Number 11 2 = 1 22 2 = 4 33 2= 9 44 2 = 16 55 2 =25 66 2 = 36 77 2 =49 88 2 =64 99 2 =81 1010 2 = 100 1111 2 = 121 1212 2 = 144 25

26. SIMPLIFYING RADICALS Find the largest perfect square. Multiply the perfect square by the other factor of the number. Put the square roots of the perfect square outside the radical symbol √. Leave the other number inside the radical. Ex. √18 = √ 9 2 = 3 √ 2 26

27. ADDING AND SUBTRACTING IRRATIONAL #S The numbers must have the same radicand (or # under the radical). Add or subtract the outside numbers and keep the radicand the same. Ex. 4 √ 3 + 5 √ 3 + √ 3 = (4+5+1) √ 3 = 10 √3 27

28. RATIONAL #S WITH DIFF. RADICANDS If the numbers have the different radicands, make the same before you add or subtract them. Ex. 5 √ 3 - √ 12 5 √ 3 - √ 4 3 5 √ 3 -√4 √ 3 5 √ 3 – 2 √ 3 = 3 √ 3 28

29. MULTIPLICATION OF IRRATIONAL #S Put both #s under one radical. Multiply the two #s. Simplify if possible. Look for perfect square numbers. Ex. 1) √ 3 √ 27 = √ 3 27 = √ 81= 9 Ex. 2) √ 6 √ 10= √ 6 10 = √ 60= √4 15 = 2 √ 15 29

30. DIVISION OF IRRATIONAL NUMBERS Put both numbers under one radical. Divide. Simplify if possible. Ex. 1) √ 8 = √8 = √4 = 2 √2 2 Ex. 2) √96 = √96 = √48 = √16 3 = √16 √3= 4√3 √2 2 30

31. RATIONALIZING THE DENOMINATOR Multiply both the numerator and the denominator by a radical. The radical on the top and bottom that you are multiplying will be the same so you are actually multiplying by the number 1. When you multiply two radicals by each other the radicals cancel out. Simplify if possible. Ex. 5 √2 = 5√2 √2 √ 2 2 31

32. APPROXIMATING SQUARE ROOTS Use your calculator! √2 (Exact value) = 1.414213562 (approximation) 32

33. PROPERTIES OF THE REAL NUMBER SYSTEM Commutative property of addition: a + b = b + a -5 + 6 = 6+ 5 -Filling your car with gasoline and washing the windshield gives the same results as washing the windshield and filling your car with gasoline Commutative property of Multiplication: a b = b a -5 6 = 6 5 Associative Property of Addition: parentheses are different -(a + b) + c = a + (b + c) -(4+2) + 6 = 4 + ( 2+ 6) -Sending a holiday card to your grandmother, sending a car to your parents, and sending a card to your teacher. 33

34. PROPERTIES OF THE REAL NUMBER SYSTEMS Associative property of multiplication: (parentheses) -(a b) c = a (b c) -Ex. (2 3) 4 = 2 (3 4) Distributive Property: multiply “the world” by what’s on the outside. -a (b + c) = a b + a c -3 ( 4+ 5) = 3 4 + 3 5 = 12 + 15 --3 ( a -4) = - 3a + 12 34

35. RULES OF EXPONENTS Evaluating the Power of a Number with Parentheses (-2) 4 = (-2)(-2)(-2)(-2) = 16 -2 4 = -1(-2)(-2)(-2)(-2)= -16 35

36. PRODUCT RULE FOR EXPONENTS When you multiply exponents, add the powers. Ex. 2 3 2 7 = 2 10 36

37. QUOTIENT RULE FOR EXPONENTS When you divide exponents, subtract the powers. 5 8 = 5 8-5= 5 3 5 5 37

38. ZERO POWER RULE FOR EXPONENTS Any number to the zero power equals 1. -Ex. 1) 2 0 = 1 -Ex. 2) (-2) 0 = 1 -Ex. 3) (5x) 0 = 1 -Ex. 4) 5x 0 = x 0 = 1 5= 5 38

39. NEGATIVE EXPONENT RULE Negative exponents go on the opposite part of the fraction (top or bottom) The sign then changes from negative to positive Ex. 2 -3 = 1 = 1 2 3 8 Ex. X -5 y -3 z 4 = r 2 s 2 z 4 q 3 r -2 s -2 q 3 x 5 y 3 39

40. POWER RULE FOR EXPONENTS When the power is outside and up high, multiply. Ex. (2 3 ) 2 = 2 32 = 2 6 40

41. SCIENTIFIC NOTATION Writing numbers to powers of ten. Convert Decimal Notation to Scientific Notation -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. -Count the number of places you have moved the decimal point to obtain the number in the first step. -Move left, exponenet is positive. -Move right, exponent is negative. -Multiply the number 10 ^ number of times you moved decimal Ex. 1) 299,000,000= 2.99 X 10 8 Ex 2) 0.000000000011= 1.1 X 10 -11 41

42. CONVERT SCIENTIFIC TO DECIMAL NOTATION Use the number on the ten. -Positive- move to the right the same number of times as the power. -Negative- move to the left the same number of times as the power. Ex. 1) 1.4 X 10 8 = 140,000,000 Ex. 2) 1.35 X 10 -3 = 0.00135 42

43. MULTIPLYING SCIENTIFIC NOTATION Multiply the numbers. Determine the exponents on the ten. (Remember when you multiply exponents, you add the powers. Ex. 1) (2.1 X 10 5 )(9 X 10 -3 ) = (2.1 X 9)(10 5 X 10 -3 ) 18.9 X 10 5-3 = 1.89 X 10 3 = 1,890 43

44. DIVIDING USING SCIENTIFIC NOTATION Write the number in scientific notation. Divide the numbers. Divide the 10’s. (Remember to subtract exponenets when you divide) Ex. 1) 0.000000000048 = 4.8 X 10 -11 = (4.8/2.4)(10 -11 /10 10 )= 24,000,000,000 2.4 X 10 10 2.0 X 10 -11-10 = 2.0 X 10 -21 44

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