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Fractions and Rational Numbers

Algebra The Easy Way – Chapter 4

Fractions and Rational Numbers
Objectives

• Introduction to fractions
• Fraction equivalence, addition, multiplication
• Reciprocals, compound fractions , fraction division
• Rules for fraction use
• Decimal fractions
• Percentages
Introduction to Fractions

• A fraction consists of two numbers with a divide sign between
– Examples: 1/2, 5/7, 2/3
– Top number: numerator; bottom number: denominator
– Denominator: how many pieces the pie is cut into
– Numerator: now many pieces you get
If numerator = denominator, result is 1 (whole pie)
• Important: denominator is not allowed to be zero
• Proper fraction: numerator < denominator
• If a, b are any two integers (b != 0), a number written in the
form a/b is called a rational number
• A fraction a/b’s reciprocal is b/a , and a/b x b/a = 1
Math Operations with Fractions

• Multiply or divide top and bottom of a fraction by the
same number, the value of the fraction stays the same

– Resulting fraction is equivalent to the original one
– However, cannot add same number to top and bottom
• Multiplying fractions: just multiply top, bottom numbers
• Adding fractions is easy as long as bottom numbers are
equal—just sum two top numbers, keep bottom number
the same

– Using pie slices, I have 2 + 3 = 5 slices of 8

• To add fractions with different denominators, multiply to
get a common denominator, then add

• Can multiply a fraction a/b by a rational number x
• Can also use this same process in reverse to remove (cancel)
portions of a fraction
• Summary of fraction rules (all denominators must not = 0):
Multiplication
Addition same denominators:
Addition, different denominators:
Simplification:
Subtraction:
Negative fractions :
Decimal Fractions

• Often difficult to compare fractions with different
denominators
– Can convert to equivalent decimal denominators, then leave
off denominator and add decimal point
– Example: 1/4 = 25/100 = 0.25
• Can convert any decimal fraction in reverse--count fraction
digits (say n), divide by 10^n
– Example: 0.354 = 354/1000
• To multiply, divide fractions: use calculator (rules are given
in the book, but faster to use “machine support”)
• Some rational numbers can’t be represented as decimal
fractions
– Example: 1/3 = 0.333333…
Percentages

• Percent: fraction with denominator = 100
– Example: 25/100 = 0.25 = 25 percent
• Percentages often used to indicate change in a quantity
• To calculate a percent change, use

– Note percent decrease is negative if new < old
– Example: old hourly pay = $10, new = $12.50, so

• If g is the percent gain, new value can be found with
new = old(1 + g/100)

Algebra The Easy Way – Chapter 5

Exponents
Objectives

• Overview of exponents
• Use of scientific notation
• Rules for exponent use
•Working with negative exponents
Overview of Exponents

• Exponents are really a shortcut way for writing “multiply a
number by itself some number of times”
– Example: (“a squared”)
– Example:
• “Three to the fourth power”
– Example: $1 at 5% for 3 years:
• 1.05 x 1.05 x 1.05 = 1.156725
• General rule for interest in year n:

• Where:
– A = initial amount,
– r = interest rate,
– n = number of years
Scientific Notation

• Another shortcut--for representing very large (or very
small) numbers
– Example: 75 trillion = 75,000,000,000,000
• Same as 7.5 x 10,000,000,000,000
• Then convert the big number to a power of
• Final result:
• Can also be written as 7.5E13
– Example: (or 2E-8)
• General rule:
– Turn number into one between 1-10
– Count zeros (to the left or right) to get “ten to the”
• Computers often use this method to represent numbers
Negative Exponents, and More Rules

• If , what is a?
– We’ll refer to a as the “ square root ” of 25 (the number, which,
when squared, equals 25)
– Answer: a = 5
– Note -5 x -5 = 25, so there are two possible answers
– When we use the radical symbol ( ), we mean the positive
square root , so
• More useful laws for manipulating exponents:
Small Fractions to Exponents; Misc. Rules

• Example: 2/100,000
•Write as
• It then turns out that (derivation on p. 67)
• So
– So negative exponent = 1/positive exponent
– Note is the reciprocal of (1/over)
• Other rules/notation:
– Anything “to the zeroth power ” = 1
• So (or =0, or undefined), etc.
– If x = 0 and n < 0, is undefined
– The symbol “±” means “plus or minus”
• Either the positive or the negative value
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