# Thread: help on complex numbers

1. ## help on complex numbers

Hello, i find this whole complex number thing quite difficult

And therefore i have 2 questions.

1
Simplify and write on the formula a + bi
e^(1-i)*e^(2+2i)

Should these functions be differentiated first or can we use the product rule and the exponetial rules to "add the exponents"?

2
Im feeling lost when dealing with e. I don't know what im doing, I don't know what it is and where it comes from. All i know is that is got something to do with exponetial growth.

2. Originally Posted by Jones
Hello, i find this whole complex number thing quite difficult

And therefore i have 2 questions.

1
Simplify and write on the formula a + bi
e^(1-i)*e^(2+2i)

Should these functions be differentiated first or can we use the product rule and the exponetial rules to "add the exponents"?
Yes the laws of exponents apply so:

e^(1-i)*e^(2+2i) = e^(1-i + 2 + 2i) = e^(3 + i)

Now to get this into the for a+bi, we need to know that:

e^(i theta) = cos(theta) + i sin(theta)

for real theta.

Now:

e^(3+i)=e^3 e^(i) = e^3 [cos(1) + i sin(1)] = e^3 cos(1) + i e^3 sin(1).

(Note trig function are in radians)

2
Im feeling lost when dealing with e. I don't know what im doing, I don't know what it is and where it comes from. All i know is that is got something to do with exponetial growth.
e is a real number a bit like pi, and e ~= 2.718282. Like pi it is important
in mathematics because it has special properties, one of which is that:

d/dx (e^x) = e^x.

RonL

3. Originally Posted by CaptainBlack

e is a real number a bit like pi, and e ~= 2.718282. Like pi it is important
in mathematics because it has special properties, one of which is that:

d/dx (e^x) = e^x.

RonL
How do we get e? Where does it come from?

4. Originally Posted by Jones
How do we get e? Where does it come from?
In several ways

e -- from Wolfram MathWorld

5. One last problem.

If we are going to write (Sqrt(3) -i) on the e^z form it should look something like this?

e^z=e^x+iy
The absolute value here is 2. So we get e^ln2, but how do i get the period y(obviously i don't mean that kind of period )

6. Originally Posted by Jones
One last problem.

If we are going to write (Sqrt(3) -i) on the e^z form it should look something like this?

e^z=e^x+iy
The absolute value here is 2. So we get e^ln2, but how do i get the period y(obviously i don't mean that kind of period )
sqrt(3) - i = 2 [sqrt(3)/2 -(1/2)i] = 2 [cos(theta) + i sin(theta)]

..............=2 e^(i theta),

so: sin(theta)=-1/2, and cos(theta)=sqrt(3)/2.

Now the first of these has solutions 7 pi/6 and 11 pi/6

(these are easier to find than you might think, first you need to know that:

sin(30 degrees) = sin( pi/6 radian) = 1/2

and then sketch the sine curve over a complete cycle 0 - 2pi to see
where the sign is what you want).

Now cos(7 pi/6) is negative which is not what we want, but cos(11pi/6)
is negative so 11pi/6 is the angle we seek.

so:

sqrt(3) - i = 2 e^(i (11pi/6)).

RonL

7. Just wanted to point out, since you moved the thread, that this is still highschool math.

And thanks for all your time and help