Expressing complex numbers in cartesian form

Then put the real and imaginary terms together (ie. real part + imaginary part). Remember to multiply the imaginary part by 'i' in order to get your final answer. For the first problem, (1+i)/(1-i), the denominator is (1-i). Its complex conjugate is (1+i). Multiply the numerator and denominator by (1+i). This gives you (1+i)^2 in the denominator which is 2i. Simplifying the numerator gives you 2i. Putting the real and imaginary parts together you get i. For the second problem, (2+3i)/(5-6i), the denominator is (5-6i). Its complex conjugate
  • #1
cmcc3119
16
2
4 Questions:

(1 + i) / (1 - i) Ans: i

(2 + 3i) / (5 - 6i) Ans: (-8+27i)/61

1/i - (3i)/(1-i) Ans: (3-5i)/2

i^123 - 4i^9 - 4^i Ans: -9i


Could someone please explain the method (detailed) as to how these answers were obatined? I understand other questions in the same field but these four I did not know how they derived the answers. Thanks for your help and your time :)
 
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  • #3
Firstly you have to convert the denominator of the complex fraction into a real number. Multiply its numerator and denominator by its complex conjugate. Remember that i^2 = -1. The complex conjugate of a+bi is a-bi. Once you have done that you only need deal with the numerator. Group all the real terms and simplify them. Do the same for the imaginary terms (ie. terms with variable 'i' in them).
 

FAQ: Expressing complex numbers in cartesian form

What is a complex number?

A complex number is a number that contains both a real part and an imaginary part. It can be written in the form a + bi, where a is the real part and bi is the imaginary part with i representing the square root of -1.

How are complex numbers expressed in cartesian form?

Complex numbers are expressed in cartesian form by breaking them down into their real and imaginary parts. The real part is represented by the x-axis on a graph, while the imaginary part is represented by the y-axis. The number is then written in the form (a, b) where a is the real part and b is the imaginary part.

What is the purpose of expressing complex numbers in cartesian form?

The purpose of expressing complex numbers in cartesian form is to make them easier to work with in mathematical operations. It allows for a visual representation of the number on a graph and simplifies calculations involving complex numbers.

Can all complex numbers be expressed in cartesian form?

Yes, all complex numbers can be expressed in cartesian form. This is because every complex number can be broken down into its real and imaginary parts, which can then be plotted on a graph.

How do you convert a complex number from polar form to cartesian form?

To convert a complex number from polar form (r, θ) to cartesian form (a, b), you can use the following formulas:
a = r * cos(θ)
b = r * sin(θ)
where r is the magnitude of the complex number and θ is the angle it makes with the positive real axis.

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