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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) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
radicals
complex
We say that c+di and c-di are complex conjugates.

2. When two complex numbers are subtracted from one another.
Complex Subtraction
cos z
Rational Number
We say that c+di and c-di are complex conjugates.

3. E ^ (z2 ln z1)
Complex Subtraction
|z-w|
Polar Coordinates - Multiplication
z1 ^ (z2)

4. A complex number and its conjugate
multiplying complex numbers
Euler's Formula
conjugate pairs
For real a and b - a + bi = 0 if and only if a = b = 0

5. No i
How to add and subtract complex numbers (2-3i)-(4+6i)
z + z*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
real

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

7. V(zz*) = v(a² + b²)
imaginary
complex numbers
|z| = mod(z)
cos iy

8. To simplify a complex fraction
z1 ^ (z2)
Imaginary number
multiply the numerator and the denominator by the complex conjugate of the denominator.
standard form of complex numbers

9. All the powers of i can be written as
'i'
Imaginary Numbers
Complex Conjugate
four different numbers: i - -i - 1 - and -1.

10. 5th. Rule of Complex Arithmetic
ln z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z-w|
Liouville's Theorem -

11. x / r
Polar Coordinates - cos?
Complex Numbers: Multiply
has a solution.
z1 ^ (z2)

12. Like pi
multiplying complex numbers
ln z
Imaginary Numbers
transcendental

13. The reals are just the
natural
a + bi for some real a and b.
Polar Coordinates - Multiplication
x-axis in the complex plane

14. z1z2* / |z2|²
Real and Imaginary Parts
z1 / z2
i^2 = -1
Complex Subtraction

15. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
sin z
Liouville's Theorem -
Roots of Unity
e^(ln z)

16. Numbers on a numberline
Affix
the vector (a -b)
Field
integers

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

18. 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
subtracting complex numbers
adding complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to find any Power

19. 1
has a solution.
i²
conjugate pairs
'i'

20. R?¹(cos? - isin?)
Polar Coordinates - z?¹
has a solution.
z1 / z2
i^0

21. x + iy = r(cos? + isin?) = re^(i?)
Square Root
Subfield
|z-w|
Polar Coordinates - z

22. 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
complex
rational
0 if and only if a = b = 0

23. Has exactly n roots by the fundamental theorem of algebra
ln z
Square Root
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)

24. We can also think of the point z= a+ ib as
Complex numbers are points in the plane
Polar Coordinates - r
Any polynomial O(xn) - (n > 0)
the vector (a -b)

25. 3rd. Rule of Complex Arithmetic
i^3
complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary Numbers

26. 1
We say that c+di and c-di are complex conjugates.
Euler Formula
i^0
standard form of complex numbers

27. 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.'
four different numbers: i - -i - 1 - and -1.
Complex Number
Complex Exponentiation
natural

28. The complex number z representing a+bi.
four different numbers: i - -i - 1 - and -1.
Affix
(cos? +isin?)n
cos iy

29. A complex number may be taken to the power of another complex number.
For real a and b - a + bi = 0 if and only if a = b = 0
sin iy
Complex Exponentiation
zz*

30. To simplify the square root of a negative number
adding complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
e^(ln z)
i^2

31. 1
Every complex number has the 'Standard Form': a + bi for some real a and b.
cosh²y - sinh²y
'i'
Integers

32. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
multiplying complex numbers
z + z*
z1 ^ (z2)

33. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
ln z
i^4
Roots of Unity

34. Written as fractions - terminating + repeating decimals
the distance from z to the origin in the complex plane
sin z
Polar Coordinates - Division
rational

35. Where the curvature of the graph changes
Euler Formula
radicals
standard form of complex numbers
point of inflection

36. Divide moduli and subtract arguments
can't get out of the complex numbers by adding (or subtracting) or multiplying two
rational
Polar Coordinates - Division
i²

37. A number that can be expressed as a fraction p/q where q is not equal to 0.
ln z
Any polynomial O(xn) - (n > 0)
Rational Number
the complex numbers

38. A subset within a field.
Subfield
i^4
the vector (a -b)
Rational Number

39. A plot of complex numbers as points.
adding complex numbers
cos iy
Argand diagram
Complex Subtraction

40. 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 Number Formula
-1
Complex Subtraction
Complex Numbers: Add & subtract

41. R^2 = x
Square Root
conjugate
a + bi for some real a and b.
i^0

42. 2a
Polar Coordinates - r
e^(ln z)
z + z*
Real and Imaginary Parts

43. When two complex numbers are added together.
Complex Addition
standard form of complex numbers
Liouville's Theorem -
Rules of Complex Arithmetic

44. The field of all rational and irrational numbers.
the distance from z to the origin in the complex plane
Every complex number has the 'Standard Form': a + bi for some real a and b.
Affix
Real Numbers

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.
How to solve (2i+3)/(9-i)
Complex Numbers: Multiply
ln z
How to find any Power

46. ? = -tan?
zz*
Euler Formula
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Arg(z*)

47. I^2 =
-1
the distance from z to the origin in the complex plane
0 if and only if a = b = 0
Affix

48. For real a and b - a + bi =
Polar Coordinates - cos?
multiplying complex numbers
0 if and only if a = b = 0
Real and Imaginary Parts

49. ½(e^(-y) +e^(y)) = cosh y
cos iy
Absolute Value of a Complex Number
sin iy
real

50. A + bi
We say that c+di and c-di are complex conjugates.
rational
Complex Exponentiation
standard form of complex numbers