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]
2 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]2 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
4y - 9 = 3 what is y
Iteru [2.4K]
4y - 9 = 3
add 9 to both sides
4y = 12
divide both sides by 4 so 4 divided by 4 is one and 12 divided by 4 is 3 thus getting your answer of 3
y=3
7 0
3 years ago
Apakah kesan kedudukan tersebut terhad
Lena [83]

ABB G Ppppwhbbbsbbsbsbsbsjsjkskenennsn

4 0
3 years ago
Please answer this correctly
brilliants [131]

Answer:

3 years 8 month

Step-by-step explanation:

Given 7 years 6 months - 3 years 10 months

Step 1: Firstly we should subtract months from months then after from year to year.

Step 2: 6 months is less than 10 months so we have to borrow 1 year and add it to the months.

Step 3: Adding 1 year to 6 months we get 18 months.

Step 4: Now we have to subtract 3 years 10 months from 6 years 18 months.

6\ years\ 18\ months-3\ years\ 10\ months=3\ years\ 8\ months

Step 5: After subtracting we get 3 years 8 months.

Thus the answer is 3 years 8 months.

8 0
3 years ago
PLEASSEE HELPPPP HURRRY FIND THE SURFACE AREA HURRRYY
Aleksandr [31]

Answer:

The answer is 48.9 cm^2.

Step-by-step explanation:

Let's find the surface areas of the faces other than the base;

triangle area: b x h / 2

4 x 7 = 28, 28/2 = 14

in this case, there are three faces, so 14 x 3 = 42, and we add that with the base to get 48.9

:) hope this helps!

8 0
2 years ago
Read 2 more answers
Convert 203.2 millimeters to inches
nata0808 [166]
So now f one in inches is 25.4 then you divide 203.2 / 25.4 = 8
3 0
3 years ago
Other questions:
  • You are leaving for a party at your cousin’s house in a city that is 150 km away. You will travel at an average rate of 50 km/hr
    14·2 answers
  • 2-7 please help BRAINLEST
    8·2 answers
  • A 6 inch personal pizza has 590 calories, with 240 of those from fat. A 16 inch pizza is cut into 8 slices. Estimate the number
    10·1 answer
  • 11x-11y=10 in slope intercept form
    14·2 answers
  • I cant figure this out!!!​
    14·1 answer
  • What is the measure of the third angle of a triangle if
    15·1 answer
  • An ice cream shop sells an empty cone for 4.00 plus an additional 1.25 per scoop of ice cream. You have 6.75 to spend. How many
    7·1 answer
  • Find the surface area of the composite figure.
    6·1 answer
  • The point (0, -4) is located.......?​
    12·2 answers
  • Solve PLS HELPPPPP PLS HELP PLS HELP PLS HELPP
    11·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!