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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. V(zz*) = v(a² + b²)
real
point of inflection
can't get out of the complex numbers by adding (or subtracting) or multiplying two
|z| = mod(z)

2. The complex number z representing a+bi.
sin z
Affix
0 if and only if a = b = 0
i^4

3. A complex number may be taken to the power of another complex number.
Complex Exponentiation
Rational Number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Square Root

4. I^2 =
Argand diagram
point of inflection
i^2
-1

5. 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.'
radicals
Polar Coordinates - r
Complex Number
natural

6. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
De Moivre's Theorem
We say that c+di and c-di are complex conjugates.
non-integers

7. 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.

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8. 3
standard form of complex numbers
i^3
Complex Addition
imaginary

9. Given (4-2i) the complex conjugate would be (4+2i)
the complex numbers
Polar Coordinates - cos?
Imaginary Numbers
Complex Conjugate

10. When two complex numbers are divided.
(cos? +isin?)n
Complex Division
Complex Multiplication
Polar Coordinates - Division

11. No i
cos z
sin iy
0 if and only if a = b = 0
real

12. 2a
Subfield
Polar Coordinates - Division
z + z*
(a + c) + ( b + d)i

13. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
i²
i^0
sin z

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

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15. 1st. Rule of Complex Arithmetic
Square Root
i²
'i'
i^2 = -1

16. x / r
Polar Coordinates - cos?
sin z
z1 ^ (z2)
adding complex numbers

17. A complex number and its conjugate
De Moivre's Theorem
sin iy
conjugate pairs
has a solution.

18. Cos n? + i sin n? (for all n integers)
Complex Number Formula
How to add and subtract complex numbers (2-3i)-(4+6i)
(cos? +isin?)n
Affix

19. Equivalent to an Imaginary Unit.
Euler Formula
Imaginary number
Polar Coordinates - Arg(z*)
natural

20. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Liouville's Theorem -
adding complex numbers
Absolute Value of a Complex Number
Rational Number

21. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Polar Coordinates - Division
radicals
Liouville's Theorem -

22. Rotates anticlockwise by p/2
has a solution.
Imaginary Numbers
Subfield
Polar Coordinates - Multiplication by i

23. 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
Real and Imaginary Parts
0 if and only if a = b = 0
integers
Rules of Complex Arithmetic

24. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
conjugate pairs
a real number: (a + bi)(a - bi) = a² + b²

25. (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
Rational Number
Complex Division
multiplying complex numbers
How to multiply complex nubers(2+i)(2i-3)

26. Have radical
z - z*
radicals
De Moivre's Theorem
(a + bi) = (c + bi) = (a + c) + ( b + d)i

27. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
z + z*
i²
the complex numbers
cos iy

28. 1
i^4
Polar Coordinates - Arg(z*)
real
non-integers

29. Numbers on a numberline
How to solve (2i+3)/(9-i)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Roots of Unity
integers

30. I
Real Numbers
Complex Addition
i^0
i^1

31. (e^(iz) - e^(-iz)) / 2i
Any polynomial O(xn) - (n > 0)
sin z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number

32. 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
Complex Numbers: Add & subtract
ln z
Affix
We say that c+di and c-di are complex conjugates.

33. ½(e^(iz) + e^(-iz))
i^1
cos z
i²
point of inflection

34. Starts at 1 - does not include 0
Field
natural
Euler's Formula
the vector (a -b)

35. ? = -tan?
cosh²y - sinh²y
Polar Coordinates - Arg(z*)
Complex Conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

36. (a + bi)(c + bi) =
e^(ln z)
Complex Number Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - sin?

37. 3rd. Rule of Complex Arithmetic
subtracting complex numbers
Complex Subtraction
Any polynomial O(xn) - (n > 0)
For real a and b - a + bi = 0 if and only if a = b = 0

38. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
How to find any Power
(a + c) + ( b + d)i
ln z
i^2 = -1

39. A subset within a field.
0 if and only if a = b = 0
sin z
Imaginary Unit
Subfield

40. A² + b² - real and non negative
Complex Multiplication
z + z*
zz*
subtracting complex numbers

41. 1
cosh²y - sinh²y
Imaginary number
imaginary
Complex numbers are points in the plane

42. E^(ln r) e^(i?) e^(2pin)
Liouville's Theorem -
e^(ln z)
Euler's Formula
Irrational Number

43. The field of all rational and irrational numbers.
Real Numbers
multiplying complex numbers
Subfield
Real and Imaginary Parts

44. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
non-integers
z1 ^ (z2)
Rational Number

45. Imaginary number

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46. 1
interchangeable
i^2
a real number: (a + bi)(a - bi) = a² + b²
Complex Subtraction

47. I
v(-1)
a + bi for some real a and b.
Rational Number
the distance from z to the origin in the complex plane

48. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
-1
Roots of Unity
cos z
Imaginary Numbers

49. 2ib
Rational Number
z - z*
cos iy
a real number: (a + bi)(a - bi) = a² + b²

50. 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.
a + bi for some real a and b.
Polar Coordinates - Division
x-axis in the complex plane