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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
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. ? = -tan?
Polar Coordinates - Arg(z*)
irrational
Complex Division
interchangeable

2. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - Division
|z-w|
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - z

3. Starts at 1 - does not include 0
natural
Integers
Polar Coordinates - z?¹
i^2 = -1

4. A+bi
Field
Complex Number Formula
zz*
can't get out of the complex numbers by adding (or subtracting) or multiplying two

5. y / r
Polar Coordinates - sin?
Integers
i^1
Complex Subtraction

6. A + bi
standard form of complex numbers
Complex Exponentiation
transcendental
real

7. Derives z = a+bi
Complex Number Formula
|z| = mod(z)
complex numbers
Euler Formula

8. To simplify a complex fraction
rational
multiply the numerator and the denominator by the complex conjugate of the denominator.
Euler's Formula
conjugate pairs

9. R^2 = x
Square Root
cosh²y - sinh²y
The Complex Numbers
cos z

10. When two complex numbers are divided.
Complex Division
Complex Addition
Affix
z - z*

11. Like pi
transcendental
Polar Coordinates - Multiplication
Polar Coordinates - sin?
natural

12. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Irrational Number
Polar Coordinates - r
Absolute Value of a Complex Number
Complex numbers are points in the plane

13. A plot of complex numbers as points.
z + z*
interchangeable
Argand diagram
cos z

14. When two complex numbers are subtracted from one another.
real
Complex Subtraction
Roots of Unity
interchangeable

15. 1
i^0
Polar Coordinates - cos?
Field
Rules of Complex Arithmetic

16. Real and imaginary numbers
Polar Coordinates - Multiplication by i
|z-w|
complex numbers
z - z*

17. 1
Polar Coordinates - r
i²
irrational
v(-1)

18. 1
Any polynomial O(xn) - (n > 0)
Real Numbers
irrational
cosh²y - sinh²y

19. ½(e^(-y) +e^(y)) = cosh y
cos iy
Complex Numbers: Add & subtract
Polar Coordinates - sin?
Complex Number

20. Rotates anticlockwise by p/2
i²
i^0
Polar Coordinates - Multiplication by i
Complex Conjugate

21. No i
0 if and only if a = b = 0
irrational
cos iy
real

22. To simplify the square root of a negative number
Field
'i'
How to multiply complex nubers(2+i)(2i-3)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

23. The modulus of the complex number z= a + ib now can be interpreted as
radicals
Complex numbers are points in the plane
Integers
the distance from z to the origin in the complex plane

24. I = imaginary unit - i² = -1 or i = v-1
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Division
Polar Coordinates - z?¹
Imaginary Numbers

25. For real a and b - a + bi =
Complex Numbers: Multiply
Affix
Complex Exponentiation
0 if and only if a = b = 0

26. 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
sin iy
subtracting complex numbers
i²
i^2

27. The complex number z representing a+bi.
Affix
Any polynomial O(xn) - (n > 0)
ln z
Subfield

28. Equivalent to an Imaginary Unit.
i^3
Imaginary number
z1 ^ (z2)
transcendental

29. Divide moduli and subtract arguments
Real Numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Division
ln z

30. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
Real and Imaginary Parts
a real number: (a + bi)(a - bi) = a² + b²
The Complex Numbers
real

31. xpressions such as ``the complex number z'' - and ``the point z'' are now
i^2
interchangeable
Irrational Number
Roots of Unity

32. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
cos z
conjugate pairs
i²

33. Any number not rational
Liouville's Theorem -
Complex Addition
a + bi for some real a and b.
irrational

34. V(x² + y²) = |z|
Complex Number
z + z*
real
Polar Coordinates - r

35. Where the curvature of the graph changes
zz*
standard form of complex numbers
point of inflection
the complex numbers

36. (a + bi) = (c + bi) =
e^(ln z)
(a + c) + ( b + d)i
Polar Coordinates - z?¹
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

37. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Division
Complex Addition

38. When two complex numbers are added together.
(a + c) + ( b + d)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
the complex numbers
Complex Addition

39. z1z2* / |z2|²
Real Numbers
How to multiply complex nubers(2+i)(2i-3)
z1 / z2
Polar Coordinates - cos?

40. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
irrational
z + z*
Complex numbers are points in the plane
ln z

41. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
i^2
conjugate
Complex Number Formula
Polar Coordinates - z?¹

42. x / r
the vector (a -b)
Affix
interchangeable
Polar Coordinates - cos?

43. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
i^1
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Division
z - z*

44. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
|z-w|
multiplying complex numbers
Complex numbers are points in the plane
has a solution.

45. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
Complex Numbers: Multiply
Liouville's Theorem -
(cos? +isin?)n
multiply the numerator and the denominator by the complex conjugate of the denominator.

46. E ^ (z2 ln z1)
complex numbers
z1 ^ (z2)
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary Numbers

47. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

48. Given (4-2i) the complex conjugate would be (4+2i)
sin iy
z + z*
multiplying complex numbers
Complex Conjugate

49. (e^(iz) - e^(-iz)) / 2i
sin z
complex
Polar Coordinates - sin?
standard form of complex numbers

50. When two complex numbers are multipiled together.
conjugate pairs
z + z*
real
Complex Multiplication