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Complex number division is also a complex number. In other terms, the divisions of 2 complex numbers can be written as A + iB, where A and B are both real numbers.
In mathematics, dividing two complex numbers produces more complex numbers. A complex number, as you may know, is made up of both real and imaginary numbers. The complex number is represented by the formula a+bi, where “a” and “b” are real numbers and i is the imaginary unit. To start dividing complex numbers, use the following formula:
Multiply the numerator and denominator of the provided complex integer by the conjugate of the denominator.
To avoid the parentheses, distribute the number in both the numerator and denominator.
Reduce the complexity of i’s abilities.
Combine similar terms and write the solution as a+bi.
If at all possible, simplify the result.
Complex Numbers
Now finding the square root of negative values is easier with complex numbers. When Hero of Alexandria, a Greek mathematician, attempted to discover the square root of a negative number in the first century, he came across the concept of complex numbers. What he all did was turn the negative into a positive and take the numeric root value. Furthermore, Italian mathematician Gerolamo Cardano defined the real identity of a complex number in the 16th century while looking for the negative roots of cubic & quadratic polynomial formulas.
Many scientific fields, such as signal processing, electromagnetism, fluid dynamics, quantum physics, & vibration analysis, use complex numbers. Here we can learn about the definition, vocabulary, visualisation, properties, and operations of complex numbers.
Division of Complex Numbers
Because it is hard to differentiate a number from an imaginary number, dividing complex numbers is a little more difficult than adding, subtracting, or multiplying them. When dividing complex numbers, we must identify a term that can be multiplied by the numerator and denominator to remove the imaginary part of denominator, resulting in a real number in the denominator.
We will learn how to divide complex numbers, divide complex numbers in polar form, divide imaginary numbers, and divide complex fractions in this post.
Because we can’t divide by an imaginary number, each fraction must have a real-number denominator, division of two complex numbers is more difficult than addition, subtraction, & multiplication.
We need to discover a term that will multiply the numerator and denominator and remove the imaginary portion of the denominator, resulting in a real integer as the denominator. The complex conjugate of the denominator is a term that is discovered by altering the sign of the imaginary component of a complex integer.
In other words, a+bi’s complex conjugate is abi. It’s worth noting that complex conjugates are reciprocal: a+bi’s complex conjugate is abi, and abi’s complex conjugate is a+bi. Complex conjugate pairs have such interesting characteristic. Their merchandise is always genuine.
What is complex number division?
Dividing complex numbers is analogous to dividing two real numbers numerically.
If z1 = x1 + iy1 & z2 = x2 + iy2 are two distinct numbers,
then dividing them is expressed as: z1/z2 = x1 + iy1/x2 + iy2.
Formula for Dividing Complex Numbers
The quotient a + ib/c + id represents the division of two complex numbers z1 = a + ib and z2 = c + id. This is determined using the z1/z2 = ac + bd/c2 + d2 + i (bc – ad / c2 + d2) formula for division of complex numbers.
Division of Complex Numbers: Steps
Now that we’ve established what dividing complex numbers entails, let’s look at the stages involved. Follow the instructions below to split the two complex numbers:
Compute the conjugate of the complex number in the fraction’s denominator first.
Multiply the conjugate by the complex fraction’s numerator and denominator.
Substitute i2 = -1 in the denominator using the algebraic identity (a+b) (a-b) = a2 – b2.
In the numerator, use the distributive property to simplify.
Now divide obtained complex number into its real and imaginary parts.
z1/z2 = (a + ib)/(c + id)
[(a + ib)/(c + id)] x [(c – id)/(c – id)]
(ac – iad + ibc – i2bd)/(c2 – (-1)d2)
[(ac + bd) + i(bc – id)]/(c2 – d2)
[(a + bd)/(c2 + d2)] + i[(bc – cd)/(c2 + d2)]
Conclusion
abi is the complex conjugate of the complex number a+bi. It is discovered by altering the sign of the complex number’s imaginary portion. The true part of the number remains the same. When you multiply a complex number by its complex conjugate, you get a real number. When we add a complex number to its complex conjugate, we get a real number.
Because we can’t divide by an imaginary number, each fraction must have a real-number denominator, division of two complex numbers is more difficult than addition, subtraction, & multiplication. We need to discover a term that will multiply the numerator and denominator and remove the imaginary portion of the denominator, resulting in a real integer as the denominator. The complex conjugate of the denominator is a term that is discovered by altering the sign of the imaginary component of a complex integer.
