  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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