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Whenever a complex number (a + bi) is multiplied by it's conjugate (a - bi), the result is a real number:

(a + bi)* (a - bi) = a2 - abi + abi - (bi)2 = a2 - b2i2 = a2 - b2(-1) = a2 + b2

This is useful when dividing complex numbers, because the numerator and denominator can both be multiplied by the denominator's conjugate, to give an equivalent fraction with a real-number denominator.

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Q: Why do you multiply by the complex conjugate?
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What is -4-3i divided by 4 plus i?

To divide by a complex number, write it as a fraction and then multiply the numerator and denominator by the complex conjugate of the denominator - this is formed by changing the sign of the imaginary bit of the number; when a complex number (a + bi) is multiplied by its complex conjugate the result is the real number a² + b² which can be divided into the complex number of the numerator: (-4 - 3i) ÷ (4 + i) = (-4 - 3i)/(4 + i) = ( (-4 - 3i)×(4 - i) ) / ( (4 + i)×(4 - i) ) = (-16 + 4i - 12i + 3i²) / (4² + 1²) = (-16 - 8i - 3) / (16 + 1) = (-19 - 8i)/17


How do you rationalise the denominator?

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If a and b are any real numbers what is the conjugate of a plus b?

The concept of conjugate is usually used in complex numbers. If your complex number is a + bi, then its conjugate is a - bi.


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