IMA101: Basic Mathematics

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LECTURE 4

IMA 101: Basic Math

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IMA101: Basic Mathematics

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Lecture Outline

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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

IMA101: Basic Mathematics

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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 …

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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

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Multiplication with decimals

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Student Practice: 67.164 * 31

46.28 * .0098

6981 * .097

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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

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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 =

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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

IMA101: Basic Mathematics

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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

IMA101: Basic Mathematics

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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

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Square roots

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Square roots

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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?

IMA101: Basic Mathematics

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Perfect Squares

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32 = 9 92=81

IMA101: Basic Mathematics

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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

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Using prime factorization√400

√2304 =

√3136=