Dealing with i - Can someone please check my reasoning? (And explain why I'm wrong if I'm wrong.)

Question

Solve 3 – 4i = x + yi

  The answers come out to be (according to the website) 3 = x and –4 = y. Below is my reasoning on how they got this.

 

For x:

If you rearrange the equation to solve for x, you get

 

x = 3 - 4i - yi which I canceled out 4i and yi because they're imaginary and the only "real" number in there that's isolated is 3.

 

For y:

If you rearrange the equation to solve for y, you get

 

y = (3 - 4i - x)/i

 

We already know x = 3 so I plug that in to get

 

y = (3 - 4i - 3)/i = -4i/i = -4 because the i's cancel out

 

I believe I did that right, but as this is m first excursion with imaginary numbers (aside from a nasty shock on a placement exam in which anything that mentioned this craziness was answered "C" because I had no clue where to start let alone end), I want to be sure I understand before proceeding.

 

Thank you in advance for your time. :)

 

~ Jennifer

Answer 1

Just for some extra clarification:

 

 

 

Jim says that a complex number is made of two parts: the real part and the imaginary part. So, if two complex numbers are equal, their real parts must be equal and their imaginary parts must be equal.

 

 

 

In 3 - 4i, the real part is 3 and the imaginary part is -4i.

 

In x + yi, the real part is x and the imaginary part is yi.

 

 

 

So, setting the real and imaginary parts equal:

 

3 = x     (1)

-4i = yi     (2)

 

From (1), you can already see that x = 3. Divide both sides of (2) by i to see that y = -4.

 

 

 

When you get more comfortable with this, you'll be able to look at any similar problem and solve it immediately just by looking at the two constants involved.

 

For example:

 

x + yi = -7 + 2i    -->    x = -7 and y = 2

x + yi = 5 - 3i    -->    x = 5 and y = -3

Answer 2

Jennifer,

          Think of a complex number as a two dimensional vector in order for two vectors to be equal their components have to be equal, the same is true for a complex number, the real and imaginary parts have to be equal.

 

the real part is x and it is equal to 3 the imaginary part is y and it is equal to -4

 

Hope this helps