Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematics of radio engineering wikipedia , lookup
Approximations of π wikipedia , lookup
Factorization wikipedia , lookup
Elementary arithmetic wikipedia , lookup
Large numbers wikipedia , lookup
Location arithmetic wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Chapter 4 Notes 4-2 Exponents Parts of an Exponent the base tells WHAT number the exponent tells HOW MANY times Example 3² means 3 x 3 = 9 −2 • a • a • a • b = −2a³b be careful of negatives (−2 )² = (−2 )(−2 ) = 4 but − 2 ² = − ( 2 • 2 ) = −4 ________________________________________________________________________________ Notes 4-7 Multiplying Exponents To multiply exponents with the same base, write the base and add the exponents. To multiply exponents with different bases, multiply the base and simplify the exponents. Taking a Power TO A POWER! This means (a²) 4 times (a²)(a²)(a²)(a²) means multiply the exponents 2 x 4 = 8 a• a• a• a• a• a• a• a = a _______________________________________________________________________________ Notes 4-8 Dividing Exponents To divide exponents with the same base, write the base and subtract the exponents. You can also end up with only a denominator. This can also be written with a negative exponent (xy) or on the bottom of a fraction. Anything to the (0) exponent = 1 2³ 2² 2¹ 2° =8 =4 =2 =1 2 =½ 2 =¼ Follow the pattern. Each answer is cut in half. Notes 5-9 Powers of products and Quotients To Raise a PRODUCT to a power - raise each factor to the power. (5 · 3)³ = 5³ · 3³ = 125 · 27 = 3375 To Raise a QUOTIENT to a power - raise both parts to the power. _______________________________________________________________________________ Notes 4-9 Scientific Notation always has this form One digit, a decimal, the rest of the number, X 10 to an exponent To make the exponent, count the spaces the decimal must move to get to decimal to its right spot. examples 6 2,000,000 = 2 X10 -5 0.00004 = 4 X10 BIG numbers have a POSITIVE exponent. Little numbers have a NEGATIVE exponent. STANDARD FORM is just a REGULAR NUMBER. ___________________________________________________________________________________________________________________ Notes 4-1 Divisibility and Factors If a number ends in 2, 4, 6, 8, or 0, it is divisible by 2. If a number ends in 5 or 0, it is divisible by 5. Example 12 ÷ 2 = 6 (no remainder) 10 ÷ 5 = 2 (no remainder) If you can add the digits of a number, and this new number is divisible by 3, 57 ÷ 3? then you can divide the original number by 3. 5 + 7 = 12 (÷ 3 = 4) 57 ÷ 3 = 19 (ALSO) If the sum of the digits is divisible by 9, you can divide the number by 9. If the sum of the digits is divisible by 3 and it is even, you can divide it by 6. Vocabulary factors Description numbers that divide evenly into another number. Example factors of 12 are 1, 2, 3, 4, 6, and 12 because 12 = 1x12, 2x6, 3x4 _______________________________________________________________________________________________________________________________________ Notes 4-3 Prime Factorization Using the Sieve of Erotosthenes we find the PRIME numbers Example 97 = 97x1 only 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 21, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97 These numbers are divisible by ONLY itself and ONE. Factor means Fracture All other numbers are COMPOSITE numbers. Use a factor tree to find the prime factorization. = 2² • 3² “Break it Down, Down, Down” Example 12 is composite (has many factors) (“Break it Down, Down, = 2² • 3² Down”) = 2² • 3² It does not matter how we start, the end numbers (prime numbers) will be the same. GCF, GCD GREATEST COMMON FACTOR, GREATEST COMMON DIVISOR (same thing) The GCF is greatest (biggest) number that is divisible into both (or more) numbers. Example 40 = 2 x 2 x 2 x 5 The factors that are the same are 2, 2, and 5 . . . 2 x 2 x 5 = 20. 60 = 2 x 2 x 3 x 5 The GCF of 40 and 60 is 20. 20 is the biggest number that is divisible into 40 and 60 evenly. ______________________________________________________________________________________________________________________________________ Notes 4-4 Simplifying Fractions Divide the numerator and denominator by the same number. All of these fractions equal ONE. To SIMPLIFY, Factor, and get rid of common (same) factors from top and bottom. Example _____________________________________________________________________________