: Complex Numbers Problems: HELP NEEDED FOR MY FINAL EXAM ?

In summary, the student is asking for help with complex numbers problems and is having difficulty understanding the lesson. They ask for help with simplifying fractions and solving equations. The student also asks if sin(a+ic)=tanb+isecb, proving that cos2acosh2c=3*cosh2.
  • #1
raladin
4
0
URGENT: Complex Numbers Problems: HELP NEEDED FOR MY FINAL EXAM!?

Q1: Write the numbers in the form a+b:

i) (2+3i)/(1+2i) - (8+i)/(6-i)

ii) [(2+i)/(6i-(1-2i))]^2


Q2: Simplify:

i) i^11

ii) i^203


Q3: Show that the points: 1, -1/2 + (i*squareroot(3))/2, -1/2 - (i*squareroot(3))/2 are the vertexes of an equilateral triangle.


Q4: Describe:

i) | 2Z - i |= 4
ii) | Z | = 3 |Z - 1|


Q5: Write in Polar Form:

i) (1+i)/[squareroot(3)-1]

ii) -2*squareroot(3) - 2i

iii) (1-i) (-squareroot(3)*i)

iv) (-1 + squareroot(3)*i)/(2+2i)


Can you please show detailed solution for each one because I don't get the lesson.. I don't have that part in my book, and it is included in my final exam after 3 days. I hope you can help me..Thanks a lot in advance!
 
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  • #2
I seriously doubt anyone will help you if you don't show your attempt.
 
  • #3
The usual method of Q1. is to multiply each fraction by the conjugate of the denominator in order to give it a real denominator... but can you please show me one solution so that I make sure?

For 2. To use the fact that i^2n = -1

For 3, To show that the distances between each point on the complex plane are equal.. but how?

On Q4, No idea

Q5 No idea
 
  • #4
raladin said:
The usual method of Q1. is to multiply each fraction by the conjugate of the denominator in order to give it a real denominator... but can you please show me one solution so that I make sure?
Again, I'm afraid that's not how the forums work. Why don't you have a go at simplifying the fractions and then we can give you some help if you make any mistakes.
 
  • #5
Whats the distance between the origin and [tex] a + ib [/tex]?
 
  • #6
The distance between the origin and a + ib = |a + bi| = sqrt(a^2 + b^2)
 
  • #7
So, what's the distance between two arbitrary points, a +ib and c +id?
 
  • #8
raladin said:
For 2. To use the fact that i^2n = -1[/B]

Not true for all N (the natural numbers i.e. 1, -1, 2, -2, etc).
What about i^4 ?
i^4 = (i^2)^2
= (-1)^2
= ?
 
  • #9


hey,in urgent need ,got exams tomorrow
can anyone help with these ques,
if sin(a+ic)=tanb+isecb,prove that cos2acosh2c=3
 

1. What are complex numbers and why are they important in mathematics?

Complex numbers are numbers that consist of 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. They are important in mathematics because they allow us to work with and solve problems involving numbers that are not on the real number line. They have many applications in fields such as physics, engineering, and computer science.

2. How do you add and subtract complex numbers?

To add or subtract complex numbers, you simply add or subtract the real parts and then add or subtract the imaginary parts. For example, (2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5)i = 6 + 8i. This is similar to combining like terms in algebra.

3. What is the difference between a complex number and a real number?

A real number is any number that can be found on the number line, including positive and negative numbers and zero. A complex number, on the other hand, is made up of a real part and an imaginary part. While all real numbers are also complex numbers, not all complex numbers are real numbers.

4. How do you multiply and divide complex numbers?

To multiply complex numbers, you use the FOIL method, just like in algebra. For example, (2 + 3i)(4 + 5i) = 2(4) + 2(5i) + 3i(4) + 3i(5i) = 8 + 10i + 12i + 15i2 = 8 + 22i - 15 = -7 + 22i. To divide complex numbers, you use the conjugate of the denominator to eliminate the imaginary part in the denominator. For example, (2 + 3i)/(4 + 5i) = ((2 + 3i)(4 - 5i))/((4 + 5i)(4 - 5i)) = (8 + 23i - 15)/(42 + 52) = (-7 + 23i)/41.

5. How can complex numbers be represented geometrically?

Complex numbers can be represented geometrically on a coordinate plane called the complex plane. The real part of the complex number is plotted on the x-axis, while the imaginary part is plotted on the y-axis. The complex number a + bi can be represented as the point (a, b) on the complex plane. The distance from the origin to the point (a, b) is the magnitude of the complex number, and the angle between the positive real axis and the line connecting the origin to (a, b) is the argument of the complex number.

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