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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. No i
real
z - z*
e^(ln z)
the complex numbers

2. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
a real number: (a + bi)(a - bi) = a² + b²
'i'
can't get out of the complex numbers by adding (or subtracting) or multiplying two

3. 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.
Rules of Complex Arithmetic
Complex Numbers: Multiply
cosh²y - sinh²y
cos z

4. 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
Rules of Complex Arithmetic
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers
has a solution.

5. Divide moduli and subtract arguments
cos z
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Numbers: Add & subtract
Polar Coordinates - Division

6. Have radical
radicals
a real number: (a + bi)(a - bi) = a² + b²
x-axis in the complex plane
Polar Coordinates - cos?

7. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
The Complex Numbers
Real Numbers
imaginary

8. Like pi
How to multiply complex nubers(2+i)(2i-3)
transcendental
z1 / z2
i^4

9. z1z2* / |z2|²
adding complex numbers
z1 / z2
Complex Number
Absolute Value of a Complex Number

10. (e^(-y) - e^(y)) / 2i = i sinh y
Complex Exponentiation
i^1
sin iy
zz*

11. 1
Subfield
Every complex number has the 'Standard Form': a + bi for some real a and b.
transcendental
i^0

12. Given (4-2i) the complex conjugate would be (4+2i)
cos iy
i²
Complex Conjugate
We say that c+di and c-di are complex conjugates.

13. Where the curvature of the graph changes
point of inflection
Complex numbers are points in the plane
|z| = mod(z)
multiplying complex numbers

14. Imaginary number

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15. V(x² + y²) = |z|
Polar Coordinates - r
Real and Imaginary Parts
Any polynomial O(xn) - (n > 0)
Polar Coordinates - cos?

16. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
four different numbers: i - -i - 1 - and -1.
Roots of Unity
Polar Coordinates - Multiplication
We say that c+di and c-di are complex conjugates.

17. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
multiplying complex numbers
Complex Number
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.
How to find any Power
Polar Coordinates - Arg(z*)
z - z*
Real and Imaginary Parts

19. E^(ln r) e^(i?) e^(2pin)
How to multiply complex nubers(2+i)(2i-3)
Integers
e^(ln z)
sin iy

20. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate pairs
cosh²y - sinh²y
sin iy

21. I
Rational Number
Complex Multiplication
i^1
Imaginary Unit

22. Starts at 1 - does not include 0
natural
z1 ^ (z2)
four different numbers: i - -i - 1 - and -1.
Affix

23. 1st. Rule of Complex Arithmetic
radicals
z1 ^ (z2)
e^(ln z)
i^2 = -1

24. A + bi
Liouville's Theorem -
has a solution.
(cos? +isin?)n
standard form of complex numbers

25. ½(e^(-y) +e^(y)) = cosh y
Every complex number has the 'Standard Form': a + bi for some real a and b.
Liouville's Theorem -
Square Root
cos iy

26. The field of all rational and irrational numbers.
Real Numbers
adding complex numbers
Polar Coordinates - z?¹
z1 / z2

27. We can also think of the point z= a+ ib as
the vector (a -b)
Absolute Value of a Complex Number
non-integers
Imaginary number

28. For real a and b - a + bi =
has a solution.
adding complex numbers
How to find any Power
0 if and only if a = b = 0

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

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30. Has exactly n roots by the fundamental theorem of algebra
i^2 = -1
adding complex numbers
subtracting complex numbers
Any polynomial O(xn) - (n > 0)

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

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32. 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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33. Every complex number has the 'Standard Form':
(a + c) + ( b + d)i
i^1
x-axis in the complex plane
a + bi for some real a and b.

34. (a + bi)(c + bi) =
Euler Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
-1
sin z

35. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Affix
z - z*
Complex Number
multiplying complex numbers

36. The reals are just the
x-axis in the complex plane
Complex Numbers: Add & subtract
Absolute Value of a Complex Number
non-integers

37. xpressions such as ``the complex number z'' - and ``the point z'' are now
Complex Division
The Complex Numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
interchangeable

38. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
Polar Coordinates - Multiplication by i
complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex numbers are points in the plane

39. A number that cannot be expressed as a fraction for any integer.
(a + c) + ( b + d)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex numbers are points in the plane
Irrational Number

40. x / r
Polar Coordinates - cos?
Square Root
i^0
Polar Coordinates - Multiplication by i

41. Derives z = a+bi
Euler Formula
non-integers
i^3
Roots of Unity

42. We see in this way that the distance between two points z and w in the complex plane is
Square Root
|z-w|
0 if and only if a = b = 0
multiply the numerator and the denominator by the complex conjugate of the denominator.

43. Equivalent to an Imaginary Unit.
Polar Coordinates - Multiplication by i
zz*
Imaginary number
natural

44. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Field
Square Root
Affix
The Complex Numbers

45. x + iy = r(cos? + isin?) = re^(i?)
i²
Integers
Roots of Unity
Polar Coordinates - z

46. 2ib
z - z*
For real a and b - a + bi = 0 if and only if a = b = 0
Irrational Number
Polar Coordinates - Arg(z*)

47. Cos n? + i sin n? (for all n integers)
How to solve (2i+3)/(9-i)
(cos? +isin?)n
Polar Coordinates - sin?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

48. All numbers
Polar Coordinates - Multiplication
complex
v(-1)
point of inflection

49. 3rd. Rule of Complex Arithmetic
z - z*
irrational
For real a and b - a + bi = 0 if and only if a = b = 0
|z-w|

50. 1
Argand diagram
Square Root
natural
i^4