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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. When two complex numbers are added together.
Complex Addition
i^4
ln z
Argand diagram

2. Starts at 1 - does not include 0
Affix
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
natural
interchangeable

3. Like pi
Polar Coordinates - z
Polar Coordinates - Division
Imaginary number
transcendental

4. The reals are just the
i^2
x-axis in the complex plane
How to multiply complex nubers(2+i)(2i-3)
the distance from z to the origin in the complex plane

5. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
i²
Complex Numbers: Add & subtract
Field
Real and Imaginary Parts

6. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - r
i^1
Complex Number

7. ? = -tan?
Argand diagram
Polar Coordinates - Arg(z*)
Polar Coordinates - Multiplication by i
|z| = mod(z)

8. z1z2* / |z2|²
four different numbers: i - -i - 1 - and -1.
i^2
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 / z2

9. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
Euler Formula
0 if and only if a = b = 0
Integers
adding complex numbers

10. y / r
Polar Coordinates - sin?
Polar Coordinates - z?¹
adding complex numbers
non-integers

11. I
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^1
Complex Conjugate
cosh²y - sinh²y

12. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

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13. Derives z = a+bi
Euler Formula
point of inflection
i^3
non-integers

14. 1
Imaginary number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
the distance from z to the origin in the complex plane
cosh²y - sinh²y

15. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

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16. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
i^2 = -1
z1 / z2
Subfield
Field

17. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Euler Formula
How to add and subtract complex numbers (2-3i)-(4+6i)
integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

18. Multiply moduli and add arguments
Polar Coordinates - Multiplication
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Subtraction
Any polynomial O(xn) - (n > 0)

19. 2nd. Rule of Complex Arithmetic

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20. Given (4-2i) the complex conjugate would be (4+2i)
Polar Coordinates - cos?
non-integers
Complex Conjugate
ln z

21. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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22. 1
z - z*
a + bi for some real a and b.
i^0
Imaginary Numbers

23. E ^ (z2 ln z1)
Argand diagram
Absolute Value of a Complex Number
Real and Imaginary Parts
z1 ^ (z2)

24. V(zz*) = v(a² + b²)
Polar Coordinates - cos?
|z| = mod(z)
z + z*
Complex Conjugate

25. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
the vector (a -b)
-1
Complex Exponentiation

26. E^(ln r) e^(i?) e^(2pin)
|z-w|
Absolute Value of a Complex Number
e^(ln z)
i^2

27. Any number not rational
|z| = mod(z)
sin z
How to solve (2i+3)/(9-i)
irrational

28. A² + b² - real and non negative
Polar Coordinates - z
z1 ^ (z2)
x-axis in the complex plane
zz*

29. Numbers on a numberline
Complex Subtraction
complex numbers
integers
complex

30. x + iy = r(cos? + isin?) = re^(i?)
Real Numbers
Polar Coordinates - z?¹
sin iy
Polar Coordinates - z

31. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
-1
zz*
Roots of Unity
Every complex number has the 'Standard Form': a + bi for some real a and b.

32. 2a
z - z*
i^2 = -1
z + z*
Field

33. All numbers
transcendental
z - z*
complex
Imaginary Numbers

34. No i
Polar Coordinates - z?¹
Roots of Unity
transcendental
real

35. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
i^0
ln z
Real and Imaginary Parts
How to solve (2i+3)/(9-i)

36. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Complex Addition
Euler Formula
z1 ^ (z2)

37. A plot of complex numbers as points.
Polar Coordinates - cos?
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Arg(z*)
Argand diagram

38. To simplify the square root of a negative number
non-integers
Every complex number has the 'Standard Form': a + bi for some real a and b.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
We say that c+di and c-di are complex conjugates.

39. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
Polar Coordinates - sin?
We say that c+di and c-di are complex conjugates.
Complex Addition
Field

40. x / r
Polar Coordinates - sin?
Square Root
v(-1)
Polar Coordinates - cos?

41. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
Polar Coordinates - r
i^2
integers
subtracting complex numbers

42. ½(e^(-y) +e^(y)) = cosh y
cos iy
The Complex Numbers
irrational
v(-1)

43. When two complex numbers are subtracted from one another.
|z-w|
Complex Subtraction
Complex Multiplication
(cos? +isin?)n

44. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Any polynomial O(xn) - (n > 0)
Affix

45. I^2 =
'i'
-1
Affix
De Moivre's Theorem

46. Where the curvature of the graph changes
point of inflection
Imaginary Unit
Subfield
(cos? +isin?)n

47. 1. i^2 = -1 2. Every complex number has the 'Standard Form': a + bi for some real a and b. 3. For real a and b - a + bi = 0 if and only if a = b = 0 4. (a + bi) = (c + bi) = (a + c) + ( b + d)i 5. (a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc
Rules of Complex Arithmetic
cos iy
De Moivre's Theorem
-1

48. A+bi
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary Numbers
Complex Number Formula
radicals

49. R?¹(cos? - isin?)
Real and Imaginary Parts
cos z
standard form of complex numbers
Polar Coordinates - z?¹

50. In this amazing number field every algebraic equation in z with complex coefficients
v(-1)
0 if and only if a = b = 0
has a solution.
We say that c+di and c-di are complex conjugates.