lecture 4 ima 101: basic math 6/17/2010 1 ima101: basic mathematics
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IMA101: Basic Mathematics
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LECTURE 4
IMA 101: Basic Math
6/17/2010
IMA101: Basic Mathematics
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Lecture Outline
6/17/2010
HW/Journal overviewWrapping up mixed numbersDecimalsSquare roots
IMA101: Basic Mathematics
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Mixed Numbers: Addition
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5
185
4
3168
5
496
3
245
IMA101: Basic Mathematics
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Mixed Numbers: Subtraction
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Remember to distribute the subtraction sign to the whole part and the fraction of the subtracted mixed number
3
2210
3
211
5
134
16
1153419
IMA101: Basic Mathematics
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Complex Fractions
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Recall: A fraction is just a number (numerator) divided by another number (denominator)
Simplify the following:
8
7
4
3
8743
83
41
21
81
311
65
31
IMA101: Basic Mathematics
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Order of Operations
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32
4
1
8
7
8
12
4
12
16
3
10
5
4
3
11
5
8
IMA101: Basic Mathematics
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Decimals
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IMA101: Basic Mathematics
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Introduction to Decimals
Decimals as fractions -1.5 8361.2759
Fractions as Decimals 5/10
3/4
1/5
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IMA101: Basic Mathematics
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Understanding Decimals
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Place valuesWriting whole numbers as decimals
42Comparing two decimals
1.99 2.99 1.999 1.99 -0.58 -0.57
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Rounding decimals
Similar to rounding whole numbers >,=5 1 <5 0
3.14159265 Round to the nearest… Tenth Hundredth Thousandth Ten-thousandth …
6/17/2010
IMA101: Basic Mathematics
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Addition and Subtraction with Decimals
Same as with whole numbersLine up the decimal points, and pull it down
to the result
Examples 382.5 – 227.1 =
2.56 – (-4.4) =
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IMA101: Basic Mathematics
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Dividing to get a decimal
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5/8 use long division
IMA101: Basic Mathematics
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Multiplication with decimals
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IGNORE the decimal pointFirst multiply by the numbers, just like you
would a whole numberThen count the number of places (in BOTH
numbers)Move the decimal of the result over that
number of placesExamples:
5.9 * 0.2 1.4 * 0.006
IMA101: Basic Mathematics
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Multiplication with decimals
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Student Practice: 67.164 * 31
46.28 * .0098
6981 * .097
IMA101: Basic Mathematics
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Decimals: Multiplication by 10
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Decimals represent fractions of 10, 100, 1000…
So multiplying a decimal by 10 means we just divide the denominator of each fraction by 10
8361.2759
Notice: we just move the decimal point to the right by 1
IMA101: Basic Mathematics
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Decimals: Multiplication by powers of 10
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Count the number of zeros and move the decimal point to the right that many spaces
3.14159265 * 10 =
3.14159265 * 100 =
3.14159265 * 1000 =
3.14159265 * 10000 =
IMA101: Basic Mathematics
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Dividing a Decimal by a whole number
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Long division: move decimal point in the same spot
71.68/ 28
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Dividing a Decimal by a Decimal
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Move decimal point over (for BOTH numbers) until you are dividing by a whole number.
0.2592/ 0.36
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Dividing by a Decimals
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Student Practice:
-5.714/ 2.4 0.02201/ 0.08
12.243 / 0.90 0.003164/0.04
IMA101: Basic Mathematics
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Dividing a Decimal by 10
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Same idea as multiplication, except this time we move the decimal point to the _____.
Example 9.0 / 10 =
4592.13 / 10 =
IMA101: Basic Mathematics
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Fractions and Decimals: Revisited
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Writing fraction as the equivalent decimalUse long division with decimals5/8
IMA101: Basic Mathematics
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Repeating decimals
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Numbers that are never factors of a power of 10 (i.e. whose factors are not 2 or 5) do not form FINITE decimals
Use a bar to indicate repeated digitsExample:1/3
IMA101: Basic Mathematics
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Repeating Decimals
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Student examples
1/11 4/7
2/9 4/13
[Rounding repeating decimals]
IMA101: Basic Mathematics
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Fractions and Decimals
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Decide if it’s easiest to work in terms of decimals or in terms of fractions
Are the decimals easily divided by the denominator?
(3/4) * 0.88 + (1/3) * 6.60
IMA101: Basic Mathematics
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6/17/2010
Square roots
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Square roots
6/17/2010
Recall: 92 = 81
81 is a perfect square9 is the Square ROOT of 81√81 = 9Recall:
-9 * -9 = 81 so -9 is also a square root of 81 We denote this as - √81 = -9
-9 * 9 = -81, but since -9≠ 9, -81 is NOT a square We can NEVER take the square root of a negative
number. Why?
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Perfect Squares
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32 = 9 92=81
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Perfect squares: memorize these
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1 2 3 4 5 6 7 8 9 10 11 12
1 1 2 3 4 5 6 7 8 9 10 11 12
2 2 4 6 8 10 12 14 16 18 20 22 24
3 3 6 9 12 15 18 21 24 27 30 33 36
4 4 8 12 16 20 24 28 32 36 40 44 48
5 5 10 15 20 25 30 35 40 45 50 55 60
6 6 12 18 24 30 36 42 48 54 60 66 72
7 7 14 21 28 35 42 49 56 63 70 77 84
8 8 16 24 32 40 48 56 64 72 80 88 96
9 9 18 27 36 45 54 63 72 81 90 99 108
10 10 20 30 40 50 60 70 80 90 100
110 120
11 11 22 33 44 55 66 77 88 99 110 121
132
12 12 24 36 48 60 72 84 96 108 120 132 144
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Properties of Square Root
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Note that we cannot “distribute” the square root when we add or subtract two terms √ 15 = √ (4+9) ≠ √4 + √9 = 2 + 3 = 5
However, when we multiply or divide the square root 6 = 2*3 = (√4) * (√ 9) =√ (4*9) = √ (36) = 6
IMA101: Basic Mathematics
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How to find the square root of a number
6/17/2010
Using prime factorization√400
√2304 =
√3136=
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