# Why do you change the signs when subtracting complex numbers? My teacher told us to change the signs, but never said why. I've flipped through the book but can't seem to find it, and it's...

Why do you change the signs when subtracting complex numbers?

My teacher told us to change the signs, but never said why. I've flipped through the book but can't seem to find it, and it's confusing me.

i.e. (6+4) - (4-3) <--- changing from the middle sign to the right inner bracet sign. Thanks (:

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### 2 Answers

When you subtract two complex numbers, it is like subtracting two linear polynomials.

`(a+bi)-(c+di)=(a-c)+(b-d)i` . In effect, you distribute a negative 1 across the second binomial.

Your example:

`(6+4i)-(4-3i)` is equivalent to:

`(6+4i)+(-1)(4-3i)` Multiplying a complex number by a scalar is just multiplying the real part and the imaginary part by the scalar -- distribute the -1 in this case:

`(6+4i)+(-4+3i)` Now add real part to real part, imaginary part to imaginary part to get:

`(6-4)+(4+3)i` or `2+7i`

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Another way to think of this is:

`(6+4i)-(4-3i)=6+4i-4-(-3i)` Drop the parantheses

`=6+4i-4+3i` since the opposite of negative 3i is positive 3i

`=2+7i` by adding like terms; reals and imaginaries.

Okay I figured it out, but to anyone who was wondering.

Because you are subtracting two polynomials, you change the signs to the opposite sign starting at the middle sign and going right.

to solve the i.e.

(6+4i) - (4-3i)

(6+4i) + (4+3i)

6+4 = 10

4i+3i = 7i

which then goes to - 10+7i (standard form)