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Complex Numbers The equation x2 + 1 = 0 has no real number solutions since any solution must be a number whose square is -1. The equation x2 + 1 = 0 has no real number solutions since any solution must be a number whose square is -1. x 1 0 2 The equation x2 + 1 = 0 has no real number solutions since any solution must be a number whose square is -1. x 1 0 1 1 2 The equation x2 + 1 = 0 has no real number solutions since any solution must be a number whose square is -1. x 1 0 1 1 2 x 2 The equation x2 + 1 = 0 has no real number solutions since any solution must be a number whose square is -1. x 1 0 1 1 2 1 x 2 In the set of real numbers, all squares are nonnegative numbers because In the set of real numbers, all squares are nonnegative numbers because x 0 2 In the set of real numbers, all squares are nonnegative numbers because x 0 2 the product of any two positive numbers is positive In the set of real numbers, all squares are nonnegative numbers because x 0 2 the product of any two positive numbers is positive 3 3 9 In the set of real numbers, all squares are nonnegative numbers because x 0 2 the product of any two positive numbers is positive & 3 3 9 the product of any two negative numbers is positive In the set of real numbers, all squares are nonnegative numbers because x 0 2 the product of any two positive numbers is positive & 3 3 9 the product of any two negative numbers is positive 3 3 9 x 2 1 x 2 1 So we must define a number i so that x 1 2 So we must define a number i so that i 2 1 For any positive real number b b i b Write the number as a product of a real number and the number i Write the number as a product of a real number and the number i 100 Write the number as a product of a real number and the number i 100 i 100 Write the number as a product of a real number and the number i 100 i 100 10i Write the number as a product of a real number and the number i Write the number as a product of a real number and the number i 36 Write the number as a product of a real number and the number i 36 i 36 Write the number as a product of a real number and the number i 36 i 36 6i Write the number as a product of a real number and the number i Write the number as a product of a real number and the number i 2 Write the number as a product of a real number and the number i 2 i 2 Multiply 4 9 Multiply 4 9 i 2 i 3 Multiply 4 9 i 2 i 3 6i 2 Multiply 4 9 i 2 i 3 6i 2 6 Multiply 3 7 Multiply 3 7 i 3 i 7 Multiply 3 7 i 3 i 7 i 2 3 7 Multiply 3 7 i 3 i 7 i 2 3 7 21 Multiply 2 8 Multiply 2 8 i 2 i 8 Multiply 2 8 i 2 i 8 i 2 2 8 Multiply 2 8 i 2 i 8 i 2 2 8 16 Multiply 2 8 i 2 i 8 i 2 2 8 16 4 Multiply 5 6 Multiply 5 6 i 5 6 Multiply 5 6 i 5 i 30 6 Divide 75 3 Divide 75 3 i 75 i 3 Divide 75 3 i 75 75 3 i 3 75 Divide 3 75 3 i 75 75 3 i 3 75 Divide 3 75 3 i 75 75 3 i 3 25 75 Divide 3 75 3 i 75 75 3 i 3 25 5 Divide 32 8 Divide i 32 32 8 8 Divide i 32 32 8 8 i 32 8 Divide i 32 32 8 8 i 32 8 i 4 Divide i 32 32 8 8 i 32 8 i 4 2i Add 2 3i 6 4i Add 2 3i 6 4i 2 3i Add 2 3i 6 4i 2 3i 6 Add 2 3i 6 4i 2 3i 6 4i Add 2 3i 6 4i 2 3i 6 4i Add 2 3i 6 4i 2 3i 6 4i 8 Add 2 3i 6 4i 2 3i 6 4i 8 7i Add 4 2i 3 i Add 4 2i 3 i 4 2i Add 4 2i 3 i 4 2i 3 Add 4 2i 3 i 4 2i 3 1i Add 4 2i 3 i 4 2i 3 1i 7 Add 4 2i 3 i 4 2i 3 1i 7 1i Add 4 2i 3 i 4 2i 3 1i 7 1i 7i Subtract 6 5i 3 2i Subtract 6 5i 3 2i 6 5i Subtract 6 5i 3 2i 6 5i Subtract 6 5i 3 2i 6 5i Subtract 6 5i 3 2i 6 5i Subtract 6 5i 3 2i 6 5i 3 Subtract 6 5i 3 2i 6 5i 3 2i Subtract 6 5i 3 2i 6 5i 3 2i Subtract 6 5i 3 2i 6 5i 3 2i 3 Subtract 6 5i 3 2i 6 5i 3 2i 3 3i Subtract 7 3i 8 6i Subtract 7 3i 8 6i 7 3i Subtract 7 3i 8 6i 7 3i Subtract 7 3i 8 6i 7 3i Subtract 7 3i 8 6i 7 3i Subtract 7 3i 8 6i 7 3i 8 6i Subtract 7 3i 8 6i 7 3i 8 6i Subtract 7 3i 8 6i 7 3i 8 6i 1 Subtract 7 3i 8 6i 7 3i 8 6i 1 3i Subtract 9 4i 9 8i Subtract 9 4i 9 8i 9 4i Subtract 9 4i 9 8i 9 4i Subtract 9 4i 9 8i 9 4i Subtract 9 4i 9 8i 9 4i Subtract 9 4i 9 8i 9 4i 9 8i Subtract 9 4i 9 8i 9 4i 9 8i Subtract 9 4i 9 8i 9 4i 9 8i 0 Subtract 9 4i 9 8i 9 4i 9 8i 0 4i Subtract 9 4i 9 8i 9 4i 9 8i 0 4i 4i Multiply 4i2 3i Multiply 4i2 3i 4i 2 Multiply 4i2 3i 4i 2 4i 3i Multiply 4i2 3i 4i 2 4i 3i 8i Multiply 4i2 3i 4i 2 4i 3i 8i 12 i 2 Multiply 4i2 3i 4i 2 4i 3i 8i 8i 12 i 2 Multiply 4i2 3i 4i 2 4i 3i 8i 12 i 8i 12 2 Multiply 4i2 3i 4i 2 4i 3i 8i 12 i 8i 12 2 a bi Multiply 4i2 3i 4i 2 4i 3i 8i 12 i 8i 12 2 a bi 12 8i Multiply 3 5i 4 2i Multiply 3 5i 4 2i 3 5i Multiply 3 5i 4 2i 3 4 2i 5i Multiply 3 5i 4 2i 3 4 2i 5i 4 2i Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i 12 Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i 12 6i Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i 12 6i 20i Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i 12 6i 20i 10i 2 Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i 12 6i 12 20i 10i 2 Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i 12 6i 12 10 20i 10i 2 Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i 12 6i 12 10 20i 10i 20i 2 Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i 12 6i 12 10 20i 10i 20i 6i 2 Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i 12 6i 12 10 22 20i 10i 20i 6i 2 Multiply 3 5i 4 2i 3 4 2i 5i 4 2i 3 4 3 2i 5i 4 5i 2i 12 6i 12 10 22 20i 10i 20i 6i 14i 2 Multiply 2 3i 1 5i Multiply 2 3i 1 5i 2 3i Multiply 2 3i 1 5i 2 1 5i 3i Multiply 2 3i 1 5i 2 1 5i 3i 1 5i Multiply 2 3i 1 5i 2 1 5i 3i 1 5i 2 Multiply 2 3i 1 5i 2 1 5i 3i 1 5i 2 10i Multiply 2 3i 1 5i 2 1 5i 3i 1 5i 2 10i 3i Multiply 2 3i 1 5i 2 1 5i 3i 1 5i 2 10i 3i 15i 2 Multiply 2 3i 1 5i 2 1 5i 3i 1 5i 2 2 10i 3i 15i 2 Multiply 2 3i 1 5i 2 1 5i 3i 1 5i 2 2 10i 15 3i 15i 2 Multiply 2 3i 1 5i 2 1 5i 3i 1 5i 2 2 10i 15 3i 3i 15i 2 Multiply 2 3i 1 5i 2 1 5i 3i 1 5i 2 2 10i 15 3i 15i 3i 10i 2 Multiply 2 3i 1 5i 2 1 5i 3i 1 5i 2 2 10i 15 17 3i 15i 3i 10i 2 Multiply 2 3i 1 5i 2 1 5i 3i 1 5i 2 2 10i 15 17 3i 15i 3i 10i 7i 2 Simplify 8 9i 5 2i Simplify 8 9i 5 2i 8 9i 5 2i 5 2i 5 2i Simplify 8 9i 5 2i 8 9i 5 2i 5 2i 5 2i 8 5 Simplify 8 9i 5 2i 8 9i 5 2i 5 2i 5 2i 8 5 8 2i Simplify 8 9i 5 2i 8 9i 5 2i 5 2i 5 2i 8 5 8 2i 9i 5 Simplify 8 9i 5 2i 8 9i 5 2i 5 2i 5 2i 8 5 8 2i 9i 5 9i 2i Simplify 8 9i 5 2i 8 9i 5 2i 5 2i 5 2i 8 5 8 2i 9i 5 9i 2i 5 5 Simplify 8 9i 5 2i 8 9i 5 2i 5 2i 5 2i 8 5 8 2i 9i 5 9i 2i 5 5 5 2i Simplify 8 9i 5 2i 8 9i 5 2i 5 2i 5 2i 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 Simplify 8 9i 5 2i 8 9i 5 2i 5 2i 5 2i 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 45i 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 45i 18 i 2 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 45i 25 18 i 2 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 45i 25 10i 18 i 2 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 45i 25 10i 10i 18 i 2 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 45i 18 i 25 10i 4 i 10i 2 2 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 45i 18 i 25 10i 4 i 40 18 10i 2 2 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 45i 18 i 25 10i 4 i 10i 40 18 45i 16i 2 2 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 45i 18 i 25 10i 4 i 10i 40 18 45i 16i 25 2 2 8 5 8 2i 9i 5 9i 2i 5 5 5 2i 2i 5 2i 2i 40 16i 45i 18 i 25 10i 4 i 10i 40 18 45i 16i 25 4 2 2 40 18 45i 16i 25 4 40 18 45i 16i 25 4 58 40 18 45i 16i 25 4 58 29i 40 18 45i 16i 25 4 58 29i 29 40 18 45i 16i 25 4 58 29i 29 29 29 40 18 45i 16i 25 4 58 29i 29 29 2 29 40 18 45i 16i 25 4 58 29i 29 29 2 i 29 40 18 45i 16i 25 4 58 29i 29 29 2 i 29 2 i 7.7 Numbers 1-68