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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 complex number may be taken to the power of another complex number.
e^(ln z)
Complex Exponentiation
Integers
a real number: (a + bi)(a - bi) = a² + b²

2. Written as fractions - terminating + repeating decimals
rational
Complex numbers are points in the plane
multiplying complex numbers
Rules of Complex Arithmetic

3. 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.
-1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex numbers are points in the plane

4. V(zz*) = v(a² + b²)
complex numbers
Euler's Formula
|z| = mod(z)
Euler Formula

5. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17
Euler Formula
complex numbers
Complex Exponentiation
How to solve (2i+3)/(9-i)

6. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
can't get out of the complex numbers by adding (or subtracting) or multiplying two
0 if and only if a = b = 0
Polar Coordinates - cos?
conjugate

7. The field of all rational and irrational numbers.
Real Numbers
the distance from z to the origin in the complex plane
Complex Exponentiation
e^(ln z)

8. 4th. Rule of Complex Arithmetic
point of inflection
radicals
(a + bi) = (c + bi) = (a + c) + ( b + d)i
conjugate pairs

9. (e^(iz) - e^(-iz)) / 2i
sin z
i^1
Real Numbers
Polar Coordinates - r

10. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
Complex numbers are points in the plane
i^0
Complex Conjugate

11. xpressions such as ``the complex number z'' - and ``the point z'' are now
sin z
interchangeable
zz*
We say that c+di and c-di are complex conjugates.

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

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13. 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
non-integers
Polar Coordinates - z?¹
Integers

14. A² + b² - real and non negative
Polar Coordinates - r
real
i^3
zz*

15. 1
-1
i^2 = -1
i^0
Rules of Complex Arithmetic

16. In this amazing number field every algebraic equation in z with complex coefficients
Polar Coordinates - Multiplication by i
has a solution.
Complex Subtraction
How to add and subtract complex numbers (2-3i)-(4+6i)

17. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
i^4
sin z
i^2 = -1

18. A subset within a field.
a + bi for some real a and b.
Subfield
Polar Coordinates - z
Argand diagram

19. A plot of complex numbers as points.
has a solution.
x-axis in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Argand diagram

20. 1
i^2
z1 / z2
Imaginary number
How to multiply complex nubers(2+i)(2i-3)

21. 2nd. Rule of Complex Arithmetic

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22. 1
Irrational Number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^3
i^4

23. Every complex number has the 'Standard Form':
Polar Coordinates - z
Complex Exponentiation
a + bi for some real a and b.
e^(ln z)

24. When two complex numbers are subtracted from one another.
Real and Imaginary Parts
How to find any Power
Complex Subtraction
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

25. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Complex Number Formula
a real number: (a + bi)(a - bi) = a² + b²
standard form of complex numbers
ln z

26. Equivalent to an Imaginary Unit.
Roots of Unity
Polar Coordinates - Division
Imaginary number
Real Numbers

27. We can also think of the point z= a+ ib as
Absolute Value of a Complex Number
Roots of Unity
the vector (a -b)
Polar Coordinates - z

28. Has exactly n roots by the fundamental theorem of algebra
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)
Complex Division

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

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30. 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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31. A + bi
standard form of complex numbers
radicals
non-integers
How to solve (2i+3)/(9-i)

32. (a + bi)(c + bi) =
z + z*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos iy
i^1

33. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
zz*
De Moivre's Theorem

34. No i
How to solve (2i+3)/(9-i)
real
Polar Coordinates - r
Complex Number

35. 2ib
z - z*
conjugate
sin z
Affix

36. A number that cannot be expressed as a fraction for any integer.
-1
Irrational Number
Field
Complex numbers are points in the plane

37. I
i^2 = -1
Imaginary Numbers
v(-1)
the vector (a -b)

38. Numbers on a numberline
z + z*
Rational Number
Square Root
integers

39. When two complex numbers are multipiled together.
Imaginary Unit
Complex Multiplication
Roots of Unity
Polar Coordinates - Multiplication

40. When two complex numbers are divided.
Any polynomial O(xn) - (n > 0)
Complex Division
zz*
0 if and only if a = b = 0

41. 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
i^4
Square Root
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Rules of Complex Arithmetic

42. 2a
How to add and subtract complex numbers (2-3i)-(4+6i)
Euler Formula
z + z*
Complex Conjugate

43. Starts at 1 - does not include 0
Polar Coordinates - Division
natural
Square Root
Polar Coordinates - Arg(z*)

44. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
i^2 = -1
z1 / z2
i^0

45. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Imaginary number
rational
0 if and only if a = b = 0

46. I^2 =
Argand diagram
sin z
ln z
-1

47. All numbers
complex
Polar Coordinates - sin?
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex numbers are points in the plane

48. x / r
Complex Multiplication
Polar Coordinates - cos?
How to solve (2i+3)/(9-i)
0 if and only if a = b = 0

49. Cos n? + i sin n? (for all n integers)
i^4
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Euler's Formula
(cos? +isin?)n

50. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Complex numbers are points in the plane
x-axis in the complex plane
complex
Field