1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
aleksandrvk [35]
3 years ago
10

Prove that if $w,z$ are complex numbers such that $|w|=|z|=1$ and $wz\ne -1$, then $\frac{w+z}{1+wz}$ is a real number.

Mathematics
1 answer:
mr_godi [17]3 years ago
3 0

Answer:

See proof below

Step-by-step explanation:

Let r=\frac{w+z}{1+wz}. If w=-z, then r=0 and r is real. Suppose that w≠-z, that is, r≠0.

Remember this useful identity: if x is a complex number then x\bar{x}=|x|^2 where \bar{x} is the conjugate of x.

Now, using the properties of the conjugate (the conjugate of the sum(product) of two numbers is the sum(product) of the conjugates):

\frac{r}{\bar{r}}=\frac{w+z}{1+wz} \left(\frac{1+\bar{w}\bar{z}}{\bar{w}+\bar{z}}{\right)

=\frac{(w+z)(1+\bar{w}\bar{z})}{(1+wz)(\bar{w}+\bar{z})}=\frac{w+z+w\bar{w}\bar{z}+z\bar{z}\bar{w}}{\bar{w}+\bar{z}+\bar{w}wz+\bar{z}zw}=\frac{w+z+w+|w|^2\bar{z}+|z|^2\bar{w}}{\bar{w}+\bar{z}+|w|^2z+|z|^2w}=\frac{w+z+\bar{z}+\bar{w}}{\bar{w}+\bar{z}+z+w}=1

Thus \frac{r}{\bar{r}}=1. From this, r=\bar{r}. A complex number is real if and only if it is equal to its conjugate, therefore r is real.

You might be interested in
Match the graph to the correct inequality.
Trava [24]
The answer to the question is b
8 0
3 years ago
Read 2 more answers
The center of a circle is at the onigin on a coordinate grd. The vertex of a parabola that opens upward is at (0, 9). If the
kenny6666 [7]
I think it’s the maximum number of solutions is one
4 0
3 years ago
HELP!!!! Which system of Linear inequalities is graphed?
wariber [46]
It is the tyhird one 
5 0
3 years ago
Read 2 more answers
Pls help asap!!!! Explain how you got your answer
bagirrra123 [75]
3+2•6= 15 because 6•2=12 and 12+3=15 so you do 15/3 which equals 5 because you divide 15 by 3
3 0
3 years ago
The ratio of 1/2 to 3 is equal to the ratio of five to what number
musickatia [10]
The ratio of 1/2 to 3 is equal to 5/6 to five
6 0
3 years ago
Other questions:
  • A regulation baseball field measures 90 feet between each base. What is the distance around the bases in yards?
    14·1 answer
  • NEED HELP PLEASE!!!!!! Jeremy surveyed students in his class about their spending habits in the school cafeteria. He used the da
    14·2 answers
  • a tree casts a shadow 32 feet long .the Angle of elevation of the sun is 62° what is the height of the tree
    15·2 answers
  • The first triangle is dilated to form the second triangle.
    6·1 answer
  • Write 98 as the product of prime factors in ascending order
    15·1 answer
  • I really suck at math plz help me
    5·2 answers
  • 5. What is x in the diagram?
    14·2 answers
  • A pizza contains 8 slices. Isabella and her friends want to order pizza. There is a total of six people eating. How many pizzas
    13·2 answers
  • Radious is given by the function c(R)=5.25r+125
    10·1 answer
  • A tree in Kira's backyard is 1/4 meters tall. It has been growing for 2 4/5 years.
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!