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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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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?
It depends on what the denominator was to start with: a surd or irrational or a complex number. You need to find the conjugate and multiply the numerator by this conjugate as well as the denominator by the conjugate. Since multiplication is by [conjugate over conjugate], which equals 1, the value is not affected. If a and b are rational numbers, then conjugate of sqrt(b) = sqrt(b) conjugate of a + sqrt(b) = a - sqrt(b), and conjugate of a + ib = a - ib where i is the imaginary square root of -1.
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.
What 2 numbers multiply to make 249 and add to 7?
The complex conjugate pair3.5 - 15.387iand3.5 + 15.387iwhere i is the imaginary square root of -1.
What is the conjugate of 11 plus 5i?
To find the complex conjugate change the sign of the imaginary part: For 11 + 5i the complex conjugate is 11 - 5i.
