Perform the operation and write all work in standard form.

Question

2/ 1 + i + 3/ 1 - i

Answer 1

2/(1+i) + 3/(1-i)

you need a common denominator which is (1+i)(1-i)

2(1-i)/[(1+i)(1-i)] + 3(1+i)/[(1-i)(1+i)]

(2-2i)/(1-i^2) + (3+3i)/(1-i^2)

(2+3-2i+3i)/[1-(-1)] because i^2=-1

(5+i)/2

(5/2)+(i/2)

(5/2)+(1/2)i

Answer 2

Hi, Serenity!

 

First of all, let me make sure I understand your question:  You are trying to add two fractions, 2/(1+i) and 3/(1-i).

 

To begin this question, the first step is the same as any other time when you are adding or subtracting two or more fractions: get a common denominator.  In this case, the two denominators that we are trying to find a common denominator with are 1+i and 1-i.  The easiest way to find a common denominator is simply to multiply all the denominators together (NOTE: This may not always give you the LOWEST common denominator, but it will always give you a common denominator.)

 

1. Finding a common denominator:

(1+i)*(1-i) = 1-i+i-i² = 1-i²

First   1*1 = 1

Outer 1*-i = -i

Inner  i*1 = i

Last    i*-i = -i²

 

2. Update the fractions with your common denominator:

   2   becomes    2*(1-i)   which simplifies to  2-2i  

(1+i)              (1+i)*(1-i)                            1-i²

 

  3    becomes     3*(1+i)   which simplifies to  3+3i 
(1-i)               (1-i)*(1+i)                              1-i²

 

3. Put it all together, and follow the rules for adding fractions with common denominators (add the numerators, keep the denominator the same).

2-2i + 3+3i = 2-2i+3+3i = 5+i

1-i²      1-i²        1-i²          1-i²

 

4. Simplify, where possible.  In this case, the fraction itself doesn't reduce to anything simpler, but i² can be rewritten as -1.  (Recall: i=√(-1), so i²=√(-1) * √(-1) = -1).  Our answer now becomes:

  5+i   = 5+i  = 5+i

1-(-1)     1+1     2

 

5. Last step: Write in standard form.  With complex numbers (anything involving an "i"), standard form is written as a+bi (so, a number without an "i" plus a number multiplied by "i").  In this case, we need to do a little more work with our answer to get it in standard form:

5+i = 5 +  i  = 5 + 1 *i

 2      2     2    2     2

(so a = 5/2 = 2.5 and b = 1/2 = 0.5)

 

I hope this answer helps!!

 

~Patty