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Sec. 2.5 Complex Numbers First, try these… Evaluate: 3 i 1 i 1 i 2 4 2 2 i 1 i i i 1 Definition: Complex Number A complex number is any number that can be written in the form a bi where a and b are real numbers. The real number a is the real part, the real number b is the imaginary part, and a + bi is the standard form. Definition: Complex Number a bi If b = 0, we have the real number a… All real numbers are also complex numbers!!! If a = 0, the number a bi becomes bi Imaginary Number Two complex numbers are equal if and only if their real and imaginary parts are equal. Operations with Complex Numbers If a + bi and c + di are two complex numbers, then Sum: a bi c di a c b d i Difference: a bi c di a c b d i Let’s Practice!!! Write the given expressions in the form bi, where b is a real number. 81 81 1 9 1 9i 20 20 1 2 5 1 2 5i Let’s Practice!!! Find the real numbers x and y that make the given equation true. x yi 2 5i x = 2, y = 5 Let’s Practice!!! Write the sum or difference in standard form. 7 3i 4 5i 11 2i 2 i 8 3i 6 4i Let’s Practice!!! Write the product in standard form. 2 3i 5 i 10 2i 15i 3i 2 Foil!!! 10 13i 3 1 13 13i Can we use our calculator to evaluate operations with complex numbers? Let’s Practice!!! If 1 3 z i , find 2 2 z 2 and z 3 . 1 3 1 3 1 3 3 3 2 z i i i i i 4 4 4 2 2 2 2 4 2 1 2 3 3 1 3 i 1 i 4 4 4 2 2 Check with calculator!!! Let’s Practice!!! 1 3 z i , find 2 2 If z 2 and z 3 . 1 3 1 3 z z z i i 2 2 2 2 3 2 1 3 3 3 2 1 3 i i i 0i 1 1 4 4 4 4 4 4 Check with calculator!!!