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Complex Numbers

Operations With Complex Numbers

Problems

Problems

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. For any complex numbers a + bı and c + dı , the following rules are true for the four basic operations.

operations

(a + bı) + (c + dı) = (a + c) + (b + d    

(a + bı) - (c + dı) = (a - c) + (b - d    

(a + bı)×(c + dı) = (ac - bd )+ (ad + bc    

fraca+bıc+dı =    

The technique for dividing complex numbers involves multiplying the denominator by its complex conjugate. The complex conjugate of a complex number a + bı is a - bı . Complex conjugates are useful and special for the following reasons:

conjugates

(a + bı) + (a - bı) = 2a    

(a + bı) - (a - bı) = 2bı    

(a + bı)×(a - bı) = a 2 + b 2    

=    

When complex conjugates are added or multiplied, the sum or product is real. So when the denominator of a fraction is a complex number, you can multiply the numerator and the denominator by the complex conjugate of the denominator to get a fraction with a real denominator.

The complex conjugate is an important tool for simplifying expressions with complex numbers. Another important fact about complex conjugates is that when a complex number is the root of a polynomial with real coefficients, so is its complex conjugate. We'll take a closer look in the next section.

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