Complex numbers

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curicuri

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Hi do someone have time to look over a task that I have problem with?

Assignment
Determine z that satisfy the equation 12458.

Solution

Using the form z=a+bi
12459
12460


My question
1) I dont get why they raise to the power of two?
2) Also do I not understand the third step when substituting for a and b?
 
Squaring essentially eliminates radicals when you expand the absolute values, [MATH]\left|a + (b-2)i\right| = \left|a + bi\right|[/MATH].

If you didn't square, then when you replace z with a + bi, you would have [MATH]\sqrt{a^2 + (b-2)^2} = \sqrt{a^2 + b^2}[/MATH], and you'd want to square next anyway.
I hope that also answers your second question.
 
curicuri said:
Assignment
Determine z that satisfy the equation View attachment 12458. Solution
Using the form z=a+bi
View attachment 12459
View attachment 12460
My question
Click to expand...
1) I dont get why they raise to the power of two?
For any numbers if \(\displaystyle |x|=|y|\) then \(\displaystyle x^2=y^2[\).
That is the reason. It is a way to simplify the question.

2) Also do I not understand the third step when substituting for a and b?
Given that \(\displaystyle z=a+bi\) then
\(\displaystyle \begin{align*}(z-2i)^2&=(a+bi-2i)^2 \\&=(a+(b-2)i)^2\\&=? \end{align*}\)


tex][/tex]tex][/tex]tex][/tex]tex][/tex]
 
Of course, I was not thinking about 12470!

But one more thing...
In the left side z appears to symbolize the constant a, but it in the rights side it symbilzes z=a+bi. Actually I think this is what confuses me the most.. When you saw this expression why did you choose to assume that z had different implications on the respective side?
 
curicuri said:
Of course, I was not thinking about View attachment 12470!

But one more thing...
In the left side z appears to symbolize the constant a, but it in the rights side it symbilzes z=a+bi. Actually I think this is what confuses me the most.. When you saw this expression why did you choose to assume that z had different implications on the respective side?
Click to expand...
z = a + bi → then

z + m = a + bi + m ................. where 'm' is a real number → then

z + m = (a + m) + bi ................. collecting real numbers together

Similarly

z + ni = a + bi + ni ................. where 'n' is a real number thus 'ni' is an imaginary number → then

z + ni = a + (b + n)i ................. collecting imaginary parts together

Similarly

z + m + ni = a + bi + m + ni ................. → then

z + m + ni = (a + m) + (b + n)i ................. collecting real and imaginary parts

Please work with pencil/paper - instead of staring at the screen. These are elementary manipulations - should be almost self-evident!!
 
curicuri said:
In the left side z appears to symbolize the constant a, but it in the rights side it symbilzes z=a+bi. Actually I think this is what confuses me the most.. When you saw this expression why did you choose to assume that z had different implications on the respective side?
Click to expand...
No, z is consistently being replaced by a + bi. To do otherwise would be wrong.

Do you see that z - 2i = (a + bi) - 2i = a + (b - 2)i?
 
Thank you for the thorough answer.
Have one more question if you dont mind.
I dont understand
12474
because when I expand (b-2)^2 from12473 I get --> b^2-4b+4. So what role do b^2 have here?
 
Here is the work they skipped:

[MATH]a^2 + (b-2)^2 = a^2 + b^2[/MATH]​
[MATH]a^2 + b^2 - 4b + 4 = a^2 + b^2[/MATH]​
[MATH]a^2 + b^2 - 4b + 4 - a^2 - b^2 = 0[/MATH]​
[MATH]-4b + 4 = 0[/MATH]​

Both a^2 and b^2 cancel. Which part did you miss? You don't show an equal sign in what you got, so maybe you didn't work with the whole equation.
[/QUOTE]
 
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