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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. 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.
a real number: (a + bi)(a - bi) = a² + b²
Complex numbers are points in the plane
Subfield
subtracting complex numbers

2. Numbers on a numberline
integers
How to find any Power
Any polynomial O(xn) - (n > 0)
i^4

3. The square root of -1.
-1
complex numbers
Polar Coordinates - cos?
Imaginary Unit

4. All the powers of i can be written as
Real Numbers
four different numbers: i - -i - 1 - and -1.
conjugate pairs
(a + c) + ( b + d)i

5. 2nd. Rule of Complex Arithmetic

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6. Equivalent to an Imaginary Unit.
the vector (a -b)
Imaginary number
Polar Coordinates - Multiplication by i
irrational

7. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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8. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
How to multiply complex nubers(2+i)(2i-3)
Every complex number has the 'Standard Form': a + bi for some real a and b.
Roots of Unity
standard form of complex numbers

9. Multiply moduli and add arguments
point of inflection
Rational Number
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication

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

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11. R^2 = x
Complex Conjugate
irrational
Square Root
Complex Subtraction

12. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Square Root
transcendental
z - z*

13. Root negative - has letter i
How to solve (2i+3)/(9-i)
Polar Coordinates - z
imaginary
Complex numbers are points in the plane

14. V(x² + y²) = |z|
Polar Coordinates - z
Polar Coordinates - r
a + bi for some real a and b.
point of inflection

15. I^2 =
-1
natural
(cos? +isin?)n
conjugate

16. Where the curvature of the graph changes
point of inflection
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - z?¹
'i'

17. V(zz*) = v(a² + b²)
multiplying complex numbers
subtracting complex numbers
irrational
|z| = mod(z)

18. A+bi
Complex Number Formula
De Moivre's Theorem
Complex numbers are points in the plane
point of inflection

19. Any number not rational
irrational
Complex Number Formula
complex numbers
Real Numbers

20. 3
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Multiply
0 if and only if a = b = 0
i^3

21. 1
Imaginary Unit
i^0
i²
We say that c+di and c-di are complex conjugates.

22. 3rd. Rule of Complex Arithmetic
Polar Coordinates - Division
|z-w|
For real a and b - a + bi = 0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

23. A subset within a field.
i^2
cos z
Subfield
Imaginary Numbers

24. y / r
i^0
(cos? +isin?)n
Polar Coordinates - sin?
Every complex number has the 'Standard Form': a + bi for some real a and b.

25. Imaginary number

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26. When two complex numbers are subtracted from one another.
zz*
Polar Coordinates - z
Complex Subtraction
multiply the numerator and the denominator by the complex conjugate of the denominator.

27. The modulus of the complex number z= a + ib now can be interpreted as
Polar Coordinates - sin?
How to multiply complex nubers(2+i)(2i-3)
subtracting complex numbers
the distance from z to the origin in the complex plane

28. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex numbers are points in the plane
Complex Division
multiplying complex numbers
Imaginary number

29. x + iy = r(cos? + isin?) = re^(i?)
Complex Exponentiation
Polar Coordinates - z
Argand diagram
imaginary

30. When two complex numbers are multipiled together.
Imaginary number
transcendental
Polar Coordinates - Division
Complex Multiplication

31. To simplify the square root of a negative number
Complex Subtraction
Complex Numbers: Multiply
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cos iy

32. Divide moduli and subtract arguments
e^(ln z)
For real a and b - a + bi = 0 if and only if a = b = 0
Field
Polar Coordinates - Division

33. z1z2* / |z2|²
z1 / z2
Liouville's Theorem -
(a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)

34. When two complex numbers are added together.
Complex Addition
'i'
0 if and only if a = b = 0
Polar Coordinates - Multiplication by i

35. Written as fractions - terminating + repeating decimals
multiplying complex numbers
Rules of Complex Arithmetic
rational
conjugate pairs

36. 5th. Rule of Complex Arithmetic
zz*
Imaginary Unit
the complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

37. All numbers
complex
a + bi for some real a and b.
Real Numbers
Complex numbers are points in the plane

38. A + bi
adding complex numbers
standard form of complex numbers
real
a + bi for some real a and b.

39. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
z1 / z2
natural
Rational Number

40. Have radical
(a + bi) = (c + bi) = (a + c) + ( b + d)i
radicals
Euler Formula
i^3

41. E ^ (z2 ln z1)
z - z*
a + bi for some real a and b.
Any polynomial O(xn) - (n > 0)
z1 ^ (z2)

42. ½(e^(iz) + e^(-iz))
cos z
i^4
Real and Imaginary Parts
natural

43. ½(e^(-y) +e^(y)) = cosh y
Rational Number
zz*
the vector (a -b)
cos iy

44. Has exactly n roots by the fundamental theorem of algebra
a + bi for some real a and b.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Any polynomial O(xn) - (n > 0)
-1

45. 1st. Rule of Complex Arithmetic
Affix
i^2 = -1
Polar Coordinates - Multiplication by i
Imaginary number

46. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
natural
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Division

47. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
i^0
real
We say that c+di and c-di are complex conjugates.
Integers

48. The complex number z representing a+bi.
Affix
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^3
can't get out of the complex numbers by adding (or subtracting) or multiplying two

49. Given (4-2i) the complex conjugate would be (4+2i)
Any polynomial O(xn) - (n > 0)
i²
Complex Conjugate
i^1

50. (a + bi)(c + bi) =
Rules of Complex Arithmetic
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
natural
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i