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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.
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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. (a + bi) = (c + bi) =
We say that c+di and c-di are complex conjugates.
(a + c) + ( b + d)i
cos z
has a solution.

2. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Rational Number
Field
radicals

3. ? = -tan?
cosh²y - sinh²y
Complex Conjugate
Polar Coordinates - Arg(z*)
non-integers

4. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
standard form of complex numbers
Euler Formula
complex numbers

5. The reals are just the
x-axis in the complex plane
Complex Exponentiation
Imaginary Numbers
non-integers

6. To simplify the square root of a negative number
a real number: (a + bi)(a - bi) = a² + b²
Complex Conjugate
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

7. Every complex number has the 'Standard Form':
a + bi for some real a and b.
Polar Coordinates - cos?
i^0
How to find any Power

8. V(x² + y²) = |z|
Every complex number has the 'Standard Form': a + bi for some real a and b.
radicals
transcendental
Polar Coordinates - r

9. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
i^4
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
real
conjugate

10. 1
transcendental
Complex Number Formula
irrational
i^2

11. (e^(-y) - e^(y)) / 2i = i sinh y
sin z
sin iy
Polar Coordinates - sin?
a + bi for some real a and b.

12. 5th. Rule of Complex Arithmetic
|z-w|
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
rational
How to multiply complex nubers(2+i)(2i-3)

13. (e^(iz) - e^(-iz)) / 2i
z + z*
z1 ^ (z2)
sin z
|z-w|

14. 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
the distance from z to the origin in the complex plane
interchangeable
subtracting complex numbers
|z-w|

15. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
the complex numbers
Complex Numbers: Multiply
How to add and subtract complex numbers (2-3i)-(4+6i)
the vector (a -b)

16. R?¹(cos? - isin?)
Polar Coordinates - z?¹
adding complex numbers
cosh²y - sinh²y
v(-1)

17. A complex number may be taken to the power of another complex number.
Complex Exponentiation
z + z*
Absolute Value of a Complex Number
Polar Coordinates - z

18. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
imaginary

19. Not on the numberline
a + bi for some real a and b.
non-integers
adding complex numbers
imaginary

20. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Polar Coordinates - sin?
Imaginary Unit
the vector (a -b)

21. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
0 if and only if a = b = 0
i^4
Field
non-integers

22. 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
'i'
the complex numbers
Polar Coordinates - cos?
How to find any Power

23. z1z2* / |z2|²
Polar Coordinates - Multiplication
z1 / z2
a real number: (a + bi)(a - bi) = a² + b²
Complex Division

24. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Complex Number
Argand diagram
Complex Exponentiation

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

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26. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
i^4
conjugate
multiply the numerator and the denominator by the complex conjugate of the denominator.

27. Like pi
x-axis in the complex plane
transcendental
Complex numbers are points in the plane
Any polynomial O(xn) - (n > 0)

28. ½(e^(-y) +e^(y)) = cosh y
point of inflection
Any polynomial O(xn) - (n > 0)
cos iy
i^3

29. Where the curvature of the graph changes
Polar Coordinates - r
has a solution.
irrational
point of inflection

30. A complex number and its conjugate
transcendental
Argand diagram
conjugate pairs
multiplying complex numbers

31. Any number not rational
the complex numbers
Polar Coordinates - Multiplication by i
irrational
cos iy

32. Starts at 1 - does not include 0
Euler's Formula
sin iy
natural
x-axis in the complex plane

33. 1st. Rule of Complex Arithmetic
i^2
rational
0 if and only if a = b = 0
i^2 = -1

34. Equivalent to an Imaginary Unit.
multiplying complex numbers
i²
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary number

35. Root negative - has letter i
We say that c+di and c-di are complex conjugates.
imaginary
How to add and subtract complex numbers (2-3i)-(4+6i)
standard form of complex numbers

36. 2a
z + z*
complex numbers
Complex Exponentiation
the vector (a -b)

37. 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
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 / z2
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

38. A² + b² - real and non negative
rational
zz*
(cos? +isin?)n
Euler Formula

39. ½(e^(iz) + e^(-iz))
x-axis in the complex plane
cos z
Imaginary Unit
sin z

40. When two complex numbers are multipiled together.
Field
complex
complex numbers
Complex Multiplication

41. A+bi
integers
Euler's Formula
Complex Number Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

42. 1
i^4
Subfield
z + z*
subtracting complex numbers

43. To simplify a complex fraction
Complex Multiplication
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to find any Power
Polar Coordinates - Division

44. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
conjugate
The Complex Numbers
i²
Polar Coordinates - cos?

45. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
can't get out of the complex numbers by adding (or subtracting) or multiplying two
complex
ln z
non-integers

46. R^2 = x
Argand diagram
subtracting complex numbers
Square Root
Complex Division

47. 1
irrational
natural
(a + c) + ( b + d)i
cosh²y - sinh²y

48. When two complex numbers are subtracted from one another.
Complex Subtraction
Polar Coordinates - sin?
Imaginary Numbers
Integers

49. 1
the distance from z to the origin in the complex plane
Complex Addition
four different numbers: i - -i - 1 - and -1.
i^0

50. Rotates anticlockwise by p/2
Complex Number Formula
Polar Coordinates - Multiplication by i
Imaginary Unit
Complex Division