Find Square Root: 5+12i Calculation

In summary, finding the square root of 5+12i involves writing the complex number in terms of two real variables and solving a system of equations. The answer is 3+2i, which can be found by setting a=2 and b=3 after solving the system.
  • #1
aisha
584
0
does anyone know how to find the square root of 5+12i ?
 
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  • #2
Try asking yourself the same question written differently:

Can you find a number z such that z*z = 5 + 12i?
 
  • #3
Hurkyl said:
Try asking yourself the same question written differently:

Can you find a number z such that z*z = 5 + 12i?

That sounds easier but I am not sure how to figure that out.

This was a multiple choice question so the answer is 2+i
I don't know how they got that!
 
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  • #4
aisha said:
This was a multiple choice question so the answer is 2+i
Actually... no, that's wrong. The answer is 3 + 2i.
 
  • #5
If it's multiple choice, then it's even easier.

You say one of the choices was 2+i. Well, what is (2+i)*(2+i)? Is it 5+12i?

Do you know of a way to write down an arbitrary complex number, z, in terms of two (real) variables? If so, then you can compute what z^2 is...
 
  • #6
phreak said:
Actually... no, that's wrong. The answer is 3 + 2i.

can u please tell me how you got that answer?
 
  • #7
remember i^2 = -1 .
 
  • #8
Well, you know that z is a complex number in the form a+bi. Therefore, you can say this:

[tex](a+bi)^2=5+12i[/tex]
[tex]a^2+2abi-b^2=5+12i[/tex]

Now, since a and b are real, you know a^2-b^2 = 5, and 2abi = 12i. Therefore, you can solve the system of equations to find a and b, thus finding z.
 
  • #9
aisha said:
can u please tell me how you got that answer?

Unfortunately, it was only a matter of guessing (and reverse factoring).

I simply used: (ai + b)^2 = 12i + 5

Expanding the equation: (ai)^2 + (ab)i + b^2 = 12i + 5

Knowing that i^2 = -1, this equation can be simplified further:

-(a^2) + 2(ab)i + b^2 = 12i + 5

so... therefore:

a*b must equal 12 and 2(b^2 - a^2) must equal 5

Make a system of equations and solve:

2ab = 12
b^2 - a^2 = 5

Solving, we find that a = 2 and b = 3

Input them into the original expanded equation (-(a^2) + 2(ab)i + b^2 = 12i + 5)

So:

-(2^2) + 2(2*3)i + 3^2 = 12i + 5

-4 + 12i + 9

12i + 5

There's an easier method to this, probably... sorry I can't really help you out.

(EDIT: This came 3 min. after nolachrymose made a post... I'm really slow at this.)
 
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  • #10
Phreak everything makes sense except for where did u get a=2 b=3? what did u solve and how? I am so dumb sorry lol I really don't like math.

Thanks Nolachrymose I would have never figured this out without u and Phreak thanks guys, :smile:
 
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  • #11
You have ab = 6, so b = 6/a
And b^2 - a^2 = 5, so 36/a^2 - a^2 = 5. This is just a quadratic in a^2. Solve it to find a^2, and hence a and b.
 

What is the formula for finding the square root of a complex number?

The formula for finding the square root of a complex number is √(a+bi) = ± (c+di), where c and d are the real and imaginary parts of the square root and are calculated using the following equations: c = √((a+√(a^2+b^2))/2) and d = ± (√((√(a^2+b^2)-a)/2)), where a and b are the real and imaginary parts of the complex number.

How do you simplify a complex number before finding its square root?

To simplify a complex number before finding its square root, you need to first determine its modulus or absolute value, which is given by √(a^2+b^2). Then, divide the real and imaginary parts of the complex number by the modulus to get a simplified form of the complex number.

How do you calculate the square root of 5+12i?

To calculate the square root of 5+12i, you first need to simplify the complex number by dividing both the real and imaginary parts by its modulus, which is √(5^2+12^2) = √(25+144) = √169 = 13. This gives us a simplified form of (5/13)+(12/13)i. Then, using the formula mentioned in the first question, we can calculate the square root as √(5+12i) = ± (√((5/13)+i(12/13))).

How do you represent the square root of a complex number on a complex plane?

The square root of a complex number can be represented on a complex plane as a point or vector that lies on the line connecting the origin (0,0) and the original complex number. The length of this point or vector is equal to the magnitude or modulus of the square root, and its direction is determined by the argument or angle of the original complex number.

How do you check the accuracy of your calculated square root of a complex number?

To check the accuracy of your calculated square root of a complex number, you can simply square the result and see if it is equal to the original complex number. If it is, then your calculated square root is correct. You can also plot the original complex number and the calculated square root on a complex plane to visually confirm their accuracy.

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