Finding the Value of a in a Complex Number Equation

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In summary: The rule for square roots is that \sqrt{ab} = \sqrt{a}\sqrt{b} provided that a and be are nonnegative. However, in this case, because a is negative, \sqrt{ab}=-1.
  • #1
thomas49th
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Homework Statement


The complex number z is defined by

[tex]z = \frac{a+2i}{a-i}[/tex]

Given that the real part of z is 1/2, find the value of a


Homework Equations


The Attempt at a Solution


Well first of all i multiplied the numerator and denominator of z by (a+i)
which gave me

[tex]\frac{a^{2} + 3ai + 1}{a^{2}-1}[/tex]

Now the real part is going to be a² + 1

so i set

[tex]\frac{a^{2} + 1}{a^{2} - 1} = \frac{1}{2}[/tex]

however I get a = [tex]\sqrt{-3}[/tex]

Have I gone the right way about solving this question?

Thanks :)
 
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  • #2
The method looks good, however I believe the numerator should be a^2 + 3ai -2, not a^2 + 3ai + 1.
 
  • #3
and the denominator should be a^2+1 not a^2-1
 
  • #4
Ahh yes i can see i made a istake in the numerator however I don't see how the denominator can be a² - 1?:
(a-i)(a+i)
= a² + ai - ai - i²
= a² -1

Does i x i always = 1 regardless of what the sign before it is?

Thanks :)
 
  • #5
thomas49th said:
Ahh yes i can see i made a istake in the numerator however I don't see how the denominator can be a² - 1?:
(a-i)(a+i)
= a² + ai - ai - i²
= a² -1

Does i x i always = 1 regardless of what the sign before it is?

Thanks :)

No … (a-i)(a+i)
= a² + ai - ai - i²
= a² + 1.

i x i always = -1 regardless of what the sign before it is! :smile:
 
  • #6
ahh i remeber asking one of my maths teachers this question

i x i is always -1

but, as surds:

[tex] \sqrt{-1} . \sqrt{-1} = \sqrt{-1 \times -1} = \sqrt{1}[/tex]

does it not?

Thanks
 
  • #7
thomas49th said:
ahh i remeber asking one of my maths teachers this question

i x i is always -1

but, as surds:

[tex] \sqrt{-1} . \sqrt{-1} = \sqrt{-1 \times -1} = \sqrt{1}[/tex]

does it not?

Thanks

hee hee! :biggrin:

nooo … it doesn't work that way …

√ (or 1/2) is ambiguous, like arcsin, and it's not a good idea to use anything ambiguous in a general formula! :smile:
 
  • #8
thomas49th said:
ahh i remeber asking one of my maths teachers this question

i x i is always -1

but, as surds:

[tex] \sqrt{-1} . \sqrt{-1} = \sqrt{-1 \times -1} = \sqrt{1}[/tex]

does it not?

Thanks
The rule for square roots is that [tex]\sqrt{ab} = \sqrt{a}\sqrt{b}[/tex] provided that a and be are nonnegative.
 
  • #9
ah cool :)
 
  • #10
hehe thomas49th I like the way you think.

While I always looked at [tex]\sqrt{-1}^2=-1[/tex] since I just cancel the square with the root, I never thought about expressing it your way. I too would've been a little confused if I ever did end up doing it that way. :smile:
 
  • #11
Mentallic said:
hehe thomas49th I like the way you think.

While I always looked at [tex]\sqrt{-1}^2=-1[/tex] since I just cancel the square with the root, I never thought about expressing it your way. I too would've been a little confused if I ever did end up doing it that way. :smile:

I hope this is the way you used to do things, but don't do them that way now! Your equation [tex]\sqrt{-1}^2=-1[/tex] is incorrect on at least two counts:
  1. [tex]-1^2=-1[/tex], so you're attempting to take the square root of -1. If you meant the square of -1, rather than the negative of 1 squared, put parentheses around -1.
  2. [tex]\sqrt{(-1)^2}=+1, not -1[/tex]
 
  • #12
Mark44 said:
I hope this is the way you used to do things, but don't do them that way now! Your equation [tex]\sqrt{-1}^2=-1[/tex] is incorrect on at least two counts:
  1. [tex]-1^2=-1[/tex], so you're attempting to take the square root of -1. If you meant the square of -1, rather than the negative of 1 squared, put parentheses around -1.
  2. [tex]\sqrt{(-1)^2}=+1, not -1[/tex]

well I meant [tex](\sqrt{-1})^2[/tex]. Of course when I see i2 I instantly think -1, and don't get into the nitty gritty of it :smile:
 

1. What are complex numbers?

Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part (with i representing the square root of -1).

2. How do you divide two complex numbers?

To divide two complex numbers, you must first rationalize the denominator by multiplying both the numerator and denominator by the complex conjugate of the denominator. Then, you can use the distributive property and simplify the resulting expression to find the quotient.

3. What is the difference between complex division and regular division?

The main difference between complex division and regular division is that in complex division, we must take into account the imaginary parts of the numbers. This means that when dividing two complex numbers, we must rationalize the denominator and use the distributive property to simplify the expression.

4. Can complex numbers be divided by real numbers?

Yes, complex numbers can be divided by real numbers. This is because real numbers can be written as complex numbers with a 0 imaginary part (a + 0i). To divide a complex number by a real number, we simply divide the real and imaginary parts separately.

5. Are there any special rules for dividing complex numbers?

Yes, there are a few special rules for dividing complex numbers. One is that the quotient of two complex numbers will always result in a complex number. Another is that when dividing by a complex number, we must rationalize the denominator. Additionally, when dividing by a real number, we can divide the real and imaginary parts separately.

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