LECTURE 4 IMA 101: Basic Math 6/17/ IMA101: Basic Mathematics

Lecture Outline 6/17/2010 IMA101: Basic Mathematics 2 HW/Journal overview Wrapping up mixed numbers Decimals Square roots

Mixed Numbers: Addition 6/17/2010 IMA101: Basic Mathematics 3

Mixed Numbers: Subtraction 6/17/2010 IMA101: Basic Mathematics 4 Remember to distribute the subtraction sign to the whole part and the fraction of the subtracted mixed number

Complex Fractions 6/17/2010 IMA101: Basic Mathematics 5 Recall: A fraction is just a number (numerator) divided by another number (denominator) Simplify the following:

Order of Operations 6/17/2010 IMA101: Basic Mathematics 6

Decimals 6/17/ IMA101: Basic Mathematics

Introduction to Decimals Decimals as fractions  -1.5  Fractions as Decimals  5/10  3/4  1/5 6/17/ IMA101: Basic Mathematics

Understanding Decimals 6/17/2010 IMA101: Basic Mathematics 9 Place values Writing whole numbers as decimals  42 Comparing two decimals   

Rounding decimals Similar to rounding whole numbers  >,=5  1  <5   Round to the nearest…  Tenth  Hundredth  Thousandth  Ten-thousandth  … 6/17/ IMA101: Basic Mathematics

Addition and Subtraction with Decimals Same as with whole numbers Line up the decimal points, and pull it down to the result Examples  – =  2.56 – (-4.4) = 6/17/ IMA101: Basic Mathematics

Dividing to get a decimal 6/17/2010 IMA101: Basic Mathematics 12 5/8  use long division

Multiplication with decimals 6/17/2010 IMA101: Basic Mathematics 13 IGNORE the decimal point First multiply by the numbers, just like you would a whole number Then count the number of places (in BOTH numbers) Move the decimal of the result over that number of places Examples:  5.9 * * 0.006

Multiplication with decimals 6/17/2010 IMA101: Basic Mathematics 14 Student Practice:  * 31  *.0098  6981 *.097

Decimals: Multiplication by 10 6/17/2010 IMA101: Basic Mathematics 15 Decimals represent fractions of 10, 100, 1000… So multiplying a decimal by 10 means we just divide the denominator of each fraction by Notice: we just move the decimal point to the right by 1

Decimals: Multiplication by powers of 10 6/17/2010 IMA101: Basic Mathematics 16 Count the number of zeros and move the decimal point to the right that many spaces * 10 = * 100 = * 1000 = * =

Dividing a Decimal by a whole number 6/17/2010 IMA101: Basic Mathematics 17 Long division: move decimal point in the same spot 71.68/ 28

Dividing a Decimal by a Decimal 6/17/2010 IMA101: Basic Mathematics 18 Move decimal point over (for BOTH numbers) until you are dividing by a whole number / 0.36

Dividing by a Decimals 6/17/2010 IMA101: Basic Mathematics 19 Student Practice: / / / /0.04

Dividing a Decimal by 10 6/17/2010 IMA101: Basic Mathematics 20 Same idea as multiplication, except this time we move the decimal point to the _____. Example  9.0 / 10 =  / 10 =

Fractions and Decimals: Revisited 6/17/2010 IMA101: Basic Mathematics 21 Writing fraction as the equivalent decimal Use long division with decimals 5/8

Repeating decimals 6/17/2010 IMA101: Basic Mathematics 22 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 digits Example: 1/3

Repeating Decimals 6/17/2010 IMA101: Basic Mathematics 23 Student examples 1/114/7 2/94/13 [Rounding repeating decimals]

Fractions and Decimals 6/17/2010 IMA101: Basic Mathematics 24 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) * (1/3) * 6.60

IMA101: Basic Mathematics 6/17/ Square roots

6/17/2010 IMA101: Basic Mathematics 26 Recall:  9 2 = is a perfect square 9 is the Square ROOT of 81 √81 = 9 Recall:  -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?

Perfect Squares 6/17/2010 IMA101: Basic Mathematics = 99 2 =81

Perfect squares: memorize these 6/17/2010 IMA101: Basic Mathematics

Properties of Square Root 6/17/2010 IMA101: Basic Mathematics 29 Note that we cannot “distribute” the square root when we add or subtract two terms  √ 15 = √ (4+9) ≠ √4 + √9 = = 5 However, when we multiply or divide the square root  6 = 2*3 = (√4) * (√ 9) =√ (4*9) = √ (36) = 6

How to find the square root of a number 6/17/2010 IMA101: Basic Mathematics 30 Using prime factorization √400 √2304 = √3136=