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IMA101 Basic Math LECTURE 2 OCTOBER 28, 2010 Lecture Outline 2 Whole numbers (continued) Multiplication Division Order of operations Integers Negative numbers Absolute value Ordered values IMA101 1/2010- Lecture 2 October 28, 2010 Multiplication (continued) 3 IMA101 1/2010- Lecture 2 October 28, 2010 Multiplication Examples 4 Times 10 Times a number with 2 digits 13 * 10 = 435 *23 = 13 * 100 = 253 * 406 13 * 1,000 = IMA101 1/2010- Lecture 1 October 26, 2010 Division 5 IMA101 1/2010- Lecture 2 October 28, 2010 Division: the opposite of multiplication 6 Take a number of items (24) and DIVIDE them into a number of groups of a certain size. How many groups are there? 24 divided by 4 = 6 4 groups of 6 IMA101 1/2010- Lecture 1 October 26, 2010 Division: Symbols 7 Example Symbol ÷ / division sign 12 ÷ 4 long division 4 12 fraction bar 12 4 1242 23 12/4 1242/23 back-slash(typed) IMA101 1/2010- Lecture 1 1242 ÷ 23 23 1242 October 26, 2010 Division: properties 8 Zero One 0 divided by any Any number divided by number is 0 0/4=0 Any number divided by zero is undefined 52 / 0 is undefined 0/0 is undefined IMA101 1/2010- Lecture 1 1 is that number 14 /1 = 14 Any nonzero number divided by itself is equal to 1 14/14 = 1 October 26, 2010 Division: Examples 9 Long division Numbers ending in zero 246 / 6 = 80 / 10 = With a remainder 169/7 = 168/5 = 40,700 / 100 = IMA101 1/2010- Lecture 1 October 26, 2010 Tests for Divisibility: A number is divisible by 10 2 if It’s last digit is divisible by 2 Example: 45,692 3 if the sum of its digits is divisible by 3 Example: 29,874 4 if it’s last 2 digits form a number divisible by 4 Example: 8,316 5 if It’s last digit is a 0 or 5 10 if It’s last digit is 0 IMA101 1/2010- Lecture 1 October 26, 2010 Division and Multiplication Table 11 11 110 2 4 6 18 3 10 24 40 5 25 8 7 60 56 96 63 4 9 12 IMA101 1/2010- Lecture 1 54 108 132 October 26, 2010 Division and Multiplication Table: Answers 12 4 9 7 3 5 11 8 6 10 2 12 11 44 99 77 33 55 121 88 66 110 22 132 2 8 18 14 6 10 22 16 12 20 4 24 6 24 54 42 18 30 66 48 36 60 12 72 3 12 27 21 9 15 33 24 18 30 6 36 10 40 90 70 30 50 110 80 60 100 20 120 5 20 45 35 15 25 55 40 30 50 10 60 8 32 72 56 24 40 88 64 48 80 16 96 7 28 63 49 21 35 77 56 42 70 14 84 4 16 36 28 12 20 44 32 24 40 8 48 9 36 81 63 27 45 99 72 54 90 18 108 12 48 108 84 36 60 132 96 72 120 24 144 IMA101 1/2010- Lecture 1 October 26, 2010 Prime Factors and Exponents 13 Factors are: Numbers multiplied together 2 * 6 = 12, so 2 and 6 are factors of 12 Prime number: A whole number that is greater than 1 and has only 1 and itself as factors 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … The only even prime number is 2 There is no pattern to prime numbers Composite number: non-prime whole numbers IMA101 1/2010- Lecture 1 October 26, 2010 Prime factorizations 14 Write 90 as a product of only prime numbers Different paths can lead to the same result What are the prime factors of 17640? IMPORTANT: Any composite number has exactly one set of prime factors IMA101 1/2010- Lecture 1 October 26, 2010 Exponents 15 Used to represent repeated factors The exponent tells us how many times a base is used as a factor Base = 2, exponent = 3 23 = 2 * 2 * 2 IMA101 1/2010- Lecture 1 October 26, 2010 Order of Operations 16 First do all calculations within parentheses: 1. 2. 3. When all grouping symbols are removed: Evaluate all powers Do multiplications and divisions as they occur from left to right Do all additions and subtractions as they occur from left to right Repeat steps 1 -3 as before If a fraction bar is present, do the operations above and below separately, then divide the number IMA101 1/2010- Lecture 1 October 26, 2010 Order of Operations: Examples 17 2(13) - 2 3 3(2 ) IMA101 1/2010- Lecture 1 October 26, 2010 Order of Operations: Examples 18 (4 2) 7 5(2 4) 7 3 IMA101 1/2010- Lecture 1 October 26, 2010 Quiz! 19 Please take out a sheet of paper and solve the following: What are the prime factors of 32760? write in exponent form (13877 2503) Solve 100 ( 5642 5462 ) 3 IMA101 1/2010- Lecture 1 October 26, 2010 Integers 20 LET’S NOT BE TOO NEGATIVE IMA101 1/2010- Lecture 2 October 28, 2010 Extending the number line 21 Positive and negative numbers Zero is neither As value of number gets higher, negative numbers get smaller Examples of negative numbers: Temperature, debt, elevation IMA101 1/2010- Lecture 2 October 28, 2010 Ordering of numbers and inequality 22 Meaning Symbol IMA101 1/2010- Lecture 2 < Less than > Greater than ≤ Less than or equal ≥ Greater than or equal ≠ Does not equal October 28, 2010 Absolute Value 23 The distance from a number to zero |-4| = |4| = 4 On the number line: IMA101 1/2010- Lecture 2 October 28, 2010 Opposite 24 The negative of a number is the point on the number line the same distance from zero but on the other side of zero On the number line: Negative of 4 is -4, and the negative of -4 is 4 IMA101 1/2010- Lecture 2 October 28, 2010 The “-” symbol 25 Can be used to show a –1 Can be used to show the –(-1) = 1 number less than zero negative of a number Can be used to subtract numbers equivalent to multiplying by negative one IMA101 1/2010- Lecture 2 or –(1) = –1 5 – 4 = 1 or 5 – (–4) = 9 5–4 = 5 + (–4) =5 + (–1*4) October 28, 2010 Integer Addition 26 If they are both positive: same as before If they are both negative: same as before and make negative Examples can be seen on the number line 4 + 6 = 10 -4 + (-6) = -10 IMA101 1/2010- Lecture 2 October 28, 2010 Integer Addition 27 If one is positive and the other is negative: Which has a smaller absolute value? Examples: 7 + (-5) If the number with the larger absolute value is positive, then the answer will be positive. On number line) Example: 5 + (-7) If the number with the larger absolute value is negative, then the answer will be negative. On number line) IMA101 1/2010- Lecture 2 October 28, 2010 Integer Subtraction 28 Analogous to adding the negative of a number Examples: 6 – 7 = 6 + (-7) = 6 – (-7) = 6 + 7 = -6 – 7 = -6 + (-7) = -6 –(-7) = -6 + 7 = IMA101 1/2010- Lecture 2 October 28, 2010