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The Set of Real Numbers Set Characteristics Examples Natural {1, 2, 3, 4, …..} (Counting) Whole {0, 1, 2, 3, 4, …..} Integers {….., -2, -1, 0, 1, 2, …..} Rational Fractions, repeating decimals, terminating decimals Irrational Non-repeating and non-terminating decimals Real 5 8 , 0.75, 0.236 1.2122....., 14 Set of all rational and irrational numbers Operations with Signed Numbers Addition 1) The sum of two positive numbers is positive. 2) The sum of two negative numbers is negative. 3) When adding a positive and a negative: QuickTime™ and a decompressor are needed to see this picture. Ignore the signs. Subtract the number with the smaller absolute value from the number with the larger absolute value. The answer has the same sign as the number with the largest absolute value. Subtraction Subtracting is the same as adding the opposite - NO DOUBLE SIGNS!! 4 - ( -3 ) = 4 + 3 Multiplication and Division 1) The product/quotient of two positives is positive. QuickTi me™ and a decompressor are needed to see thi s pi ctur e. 2) The product/quotient of two negatives is positive. 3) The product/quotient of a positive and a negative is negative. Simplifying Radical Expressions 1) Check if the radicand is a perfect square. If so, just simplify! Example 2) 36 = 6 or 25 81 5 9 If the radicand is not a perfect square, determine the largest perfect square that divides into the radicand. Example 44 4 11 Simplify the square root of the perfect square. That number becomes the coefficient of the radical. Example 44 4 11 2 11 Sometimes, it’s difficult to find the largest perfect square. To help you, you could divide the radicand by 2. Check the perfect squares less than or equal to the quotient. Starting with the larger numbers is a good way of attacking it. Example 180 is a big number!!! Where to start? 180 180 2 90 Start with squares less than 90. 81 doesn’t divide into 180, nor 64, nor 49. But, 180 36 5 6 5 3) If the radical has a coefficient, simplify the radical and multiply the coefficients. Example 5 90 5 9 10 5 3 10 15 10 The Most Important Thing: Memorize the Perfect Squares!!