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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. Where the curvature of the graph changes
the vector (a -b)
point of inflection
z + z*
complex

2. For real a and b - a + bi =
non-integers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
0 if and only if a = b = 0
the vector (a -b)

3. The modulus of the complex number z= a + ib now can be interpreted as
Polar Coordinates - Multiplication by i
Real Numbers
the distance from z to the origin in the complex plane
-1

4. 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.'
i^3
Integers
Complex Number
'i'

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

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6. In this amazing number field every algebraic equation in z with complex coefficients
real
has a solution.
Euler's Formula
i^0

7. V(x² + y²) = |z|
Irrational Number
Polar Coordinates - r
cosh²y - sinh²y
'i'

8. A subset within a field.
complex
z1 ^ (z2)
Subfield
non-integers

9. All the powers of i can be written as
(a + c) + ( b + d)i
irrational
four different numbers: i - -i - 1 - and -1.
Real and Imaginary Parts

10. ½(e^(-y) +e^(y)) = cosh y
cos iy
irrational
Imaginary Numbers
Argand diagram

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

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12. A+bi
x-axis in the complex plane
Complex Number Formula
Polar Coordinates - z
a + bi for some real a and b.

13. 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
subtracting complex numbers
Square Root
a real number: (a + bi)(a - bi) = a² + b²
z - z*

14. Imaginary number

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15. (a + bi)(c + bi) =
How to find any Power
How to multiply complex nubers(2+i)(2i-3)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable

16. When two complex numbers are added together.
complex numbers
De Moivre's Theorem
Complex Addition
(a + bi) = (c + bi) = (a + c) + ( b + d)i

17. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
the distance from z to the origin in the complex plane
radicals
Roots of Unity

18. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
-1
i^0
How to find any Power
Any polynomial O(xn) - (n > 0)

19. Derives z = a+bi
Euler Formula
ln z
a + bi for some real a and b.
Every complex number has the 'Standard Form': a + bi for some real a and b.

20. The square root of -1.
Euler's Formula
cosh²y - sinh²y
Imaginary Unit
De Moivre's Theorem

21. Equivalent to an Imaginary Unit.
Imaginary number
Argand diagram
The Complex Numbers
Real Numbers

22. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
The Complex Numbers
point of inflection
Subfield
conjugate

23. Like pi
i^2
transcendental
Euler's Formula
irrational

24. 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
sin iy
For real a and b - a + bi = 0 if and only if a = b = 0
rational

25. Not on the numberline
Subfield
Polar Coordinates - cos?
non-integers
z1 / z2

26. R?¹(cos? - isin?)
De Moivre's Theorem
sin z
Polar Coordinates - z?¹
i^2

27. To simplify a complex fraction
rational
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex
Polar Coordinates - r

28. 2nd. Rule of Complex Arithmetic

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29. 3
i^3
Irrational Number
natural
Imaginary Unit

30. 3rd. Rule of Complex Arithmetic
radicals
i^0
cos iy
For real a and b - a + bi = 0 if and only if a = b = 0

31. 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
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Division
conjugate pairs
i²

32. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
a + bi for some real a and b.
a real number: (a + bi)(a - bi) = a² + b²
i^2 = -1

33. Divide moduli and subtract arguments
Polar Coordinates - Division
Roots of Unity
How to add and subtract complex numbers (2-3i)-(4+6i)
transcendental

34. A complex number may be taken to the power of another complex number.
How to find any Power
Rational Number
Complex Exponentiation
irrational

35. I
Argand diagram
We say that c+di and c-di are complex conjugates.
imaginary
v(-1)

36. ? = -tan?
Complex Conjugate
Polar Coordinates - Division
Polar Coordinates - Arg(z*)
Every complex number has the 'Standard Form': a + bi for some real a and b.

37. Real and imaginary numbers
standard form of complex numbers
Polar Coordinates - Division
complex numbers
irrational

38. (e^(iz) - e^(-iz)) / 2i
sin z
subtracting complex numbers
a + bi for some real a and b.
Integers

39. We see in this way that the distance between two points z and w in the complex plane is
multiply the numerator and the denominator by the complex conjugate of the denominator.
|z-w|
z + z*
rational

40. E^(ln r) e^(i?) e^(2pin)
(a + c) + ( b + d)i
imaginary
Euler's Formula
e^(ln z)

41. x / r
Polar Coordinates - cos?
standard form of complex numbers
i²
Polar Coordinates - Division

42. 1
i^4
the vector (a -b)
Complex Numbers: Add & subtract
i²

43. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Polar Coordinates - Arg(z*)
z1 / z2
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Field

44. (a + bi) = (c + bi) =
conjugate
(a + c) + ( b + d)i
Polar Coordinates - Multiplication
(cos? +isin?)n

45. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
non-integers
Complex Numbers: Multiply
|z-w|

46. All numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
cos z
complex
i^2

47. When two complex numbers are subtracted from one another.
How to solve (2i+3)/(9-i)
Complex Subtraction
De Moivre's Theorem
Real and Imaginary Parts

48. I^2 =
-1
For real a and b - a + bi = 0 if and only if a = b = 0
Square Root
e^(ln z)

49. Cos n? + i sin n? (for all n integers)
Rules of Complex Arithmetic
(cos? +isin?)n
Polar Coordinates - r
Polar Coordinates - Arg(z*)

50. We can also think of the point z= a+ ib as
integers
imaginary
the vector (a -b)
Real Numbers