prove: if z is a complex number 1/2= the conjugate of z of the absolute value of z squared.
Please check problem statement for typos. I do not understand it.
prove: if z is a complex number and 1/z=the conjugate of z over the absolute value of z squared.
let z = a + b i
then z' = a - b i (using z' for conjugate notation)
1
------ *(a-bi)/(a-bi)
a+bi
= a - bi
-----------
a^2+b^2
the end :)
To prove the given statement, we need to show that for any complex number z, the equation
1/2 = |z|^2
holds true, where |z| is the absolute value of z.
Let's break down the proof step by step:
Step 1: Start with a complex number z.
Let z = a + bi, where a and b are real numbers and i is the imaginary unit.
Step 2: Calculate the absolute value of z.
The absolute value of z, denoted |z|, is equal to the square root of the sum of the squares of the real and imaginary parts of z.
|z| = √(a^2 + b^2)
Step 3: Calculate the square of the absolute value of z.
|z|^2 = (a^2 + b^2)^2
Step 4: Find the conjugate of z.
The conjugate of z is obtained by changing the sign of the imaginary part of z.
Conjugate of z = a - bi
Step 5: Calculate the product of z and its conjugate.
z * conjugate of z = (a + bi)(a - bi) = a^2 - abi + abi - b^2 * i^2
= a^2 - b^2 * (-1)
= a^2 + b^2
Step 6: Rearrange the equation to prove the statement.
1/2 = (a^2 + b^2)/(a^2 + b^2)
= |z|^2/|z|^2
= 1
Conclusion:
By following the steps above, we have shown that for any complex number z, the equation 1/2 = |z|^2 is true.