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Graphing Complex Numbers AND Finding the Absolute Value of Complex Numbers SPI 3103.2.2 Compute with all real and complex numbers. Checks for Understanding 3103.2.7 Graph complex numbers in the complex plane and recognize differences and similarities with the graphical representations of real numbers graphed on the number line. 3103.2.9 Find and describe geometrically the absolute value of a complex number. Graphing Complex Numbers • Complex numbers cannot be graphed on a normal coordinate axes. • Complex numbers are graphed in an Argand diagram, which looks very much like a regular Cartesian coordinate axes. • An Argand diagram shows a relationship between the x-axis (real axis) with real numbers and the y-axis (imaginary axis) with imaginary numbers. • In an Argand diagram, a complex number (a + bi) is the point (a, b) or the vector from the origin to the point (a, b). Argand Diagram Imaginary axis Real axis Graph 2 + 5i The graph of 2 + 5i is represented by the point (2, 5) OR by the vector from the origin to the point (2, 5). yi 2 + 5i x Graph 5 – 6i yi The graph of 5 – 6i is represented by the point (5, –6) OR by the vector from the origin to the point (5, –6). x 5 – 6i Graph 3i The graph of 3i is represented by the point (0, 3) OR by the vector from the origin to the point (0, 3). 3i is the same as 0 + 3i. yi 3i x Graph –7 The graph of –7 is represented by the point (– 7, 0) OR by the vector from the origin to the point (– 7, 0). –7 is the same as –7 + 0i yi –7 x Try These 1. –2 + 7i 2. –6 – i 3. 2 4. 8i Absolute Value of Complex Numbers • The absolute value of a real number is the distance from zero to the number on the number line. • The absolute value of a complex number is also the distance from the number to zero, but the distance is measured from zero to the number in an Argand diagram rather than on a number line. • The most efficient method to find the absolute value of a complex number is derived from the Pythagorean Theorem. Absolute Value of Complex Numbers • The absolute value of a complex number z = a + bi is written as z . • The absolute value of a complex number is a nonnegative real number defined as z = a2 b2 . • Since a complex number is represented by a point or by the vector from the origin to the point, the absolute value is the length of the vector, called the magnitude. Find the absolute value of 3 + 4i yi To find the absolute value of a complex number, find the distance from the number to the origin. The formula to find the absolute value of a complex number is as z = a2 b2 . 3 + 4i x Find the absolute value of 3 + 4i yi z = a 2 b2 3 + 4i = 3 + 4i 32 4 2 3 + 4i = 25 3 + 4i = 5 4 3 x Find the absolute value of 2 – 3i yi z = a 2 b2 2 – 3i 2 2 2 ( 3) = 2 – 3i = 4 9 2 x 3 2 – 3i = 13 2 – 3i Try These 1. –4 + 6i 2. –3 + 5i