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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. The complex number z representing a+bi.
Real Numbers
Irrational Number
Affix
i²

2. 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
imaginary
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Multiplication by i

3. Equivalent to an Imaginary Unit.
Complex Multiplication
Imaginary number
conjugate
i^4

4. z1z2* / |z2|²
Imaginary Numbers
x-axis in the complex plane
z1 / z2
complex numbers

5. ½(e^(-y) +e^(y)) = cosh y
Roots of Unity
cos iy
a real number: (a + bi)(a - bi) = a² + b²
z + z*

6. 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
zz*
z1 / z2
Irrational Number
subtracting complex numbers

7. 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
How to find any Power
Rules of Complex Arithmetic
Integers
sin iy

8. Starts at 1 - does not include 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Field
natural
cos iy

9. 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.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic
Imaginary Unit
Complex Numbers: Multiply

10. Numbers on a numberline
integers
multiply the numerator and the denominator by the complex conjugate of the denominator.
i²
can't get out of the complex numbers by adding (or subtracting) or multiplying two

11. 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.
Roots of Unity
Complex numbers are points in the plane
The Complex Numbers
|z| = mod(z)

12. I
Liouville's Theorem -
Complex Numbers: Multiply
standard form of complex numbers
v(-1)

13. A complex number may be taken to the power of another complex number.
a real number: (a + bi)(a - bi) = a² + b²
Complex Exponentiation
irrational
the distance from z to the origin in the complex plane

14. E ^ (z2 ln z1)
Every complex number has the 'Standard Form': a + bi for some real a and b.
transcendental
Polar Coordinates - z?¹
z1 ^ (z2)

15. (a + bi)(c + bi) =
complex
How to add and subtract complex numbers (2-3i)-(4+6i)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable

16. A plot of complex numbers as points.
|z-w|
(a + c) + ( b + d)i
Argand diagram
Any polynomial O(xn) - (n > 0)

17. A number that cannot be expressed as a fraction for any integer.
Complex numbers are points in the plane
Complex Conjugate
Irrational Number
i^0

18. For real a and b - a + bi =
-1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
(a + c) + ( b + d)i
0 if and only if a = b = 0

19. A complex number and its conjugate
conjugate pairs
cos iy
z1 / z2
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

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

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21. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Polar Coordinates - Multiplication
Imaginary Unit
Euler Formula

22. Rotates anticlockwise by p/2
adding complex numbers
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Multiplication by i
Complex Subtraction

23. Cos n? + i sin n? (for all n integers)
complex
Absolute Value of a Complex Number
Complex Addition
(cos? +isin?)n

24. 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
Euler Formula
Complex Numbers: Add & subtract
i^2
Liouville's Theorem -

25. R^2 = x
cosh²y - sinh²y
How to solve (2i+3)/(9-i)
Square Root
Polar Coordinates - sin?

26. A + bi
Complex Number Formula
standard form of complex numbers
Complex Exponentiation
can't get out of the complex numbers by adding (or subtracting) or multiplying two

27. R?¹(cos? - isin?)
How to solve (2i+3)/(9-i)
Polar Coordinates - z?¹
Complex Number Formula
Liouville's Theorem -

28. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z
sin iy
four different numbers: i - -i - 1 - and -1.

29. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
-1
|z-w|
i^4
The Complex Numbers

30. 1
can't get out of the complex numbers by adding (or subtracting) or multiplying two
point of inflection
0 if and only if a = b = 0
i^0

31. Root negative - has letter i
imaginary
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary number
i^2

32. When two complex numbers are divided.
Polar Coordinates - Division
Complex Division
(cos? +isin?)n
irrational

33. ½(e^(iz) + e^(-iz))
Rational Number
cos z
0 if and only if a = b = 0
ln z

34. 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
the complex numbers
i²
Complex Conjugate
conjugate pairs

35. 3rd. Rule of Complex Arithmetic
Polar Coordinates - sin?
Field
For real a and b - a + bi = 0 if and only if a = b = 0
Roots of Unity

36. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
the complex numbers
i^2 = -1
i^3

37. 1
i^3
Euler's Formula
i²
i^2

38. No i
e^(ln z)
real
Complex numbers are points in the plane
point of inflection

39. The square root of -1.
has a solution.
Real and Imaginary Parts
Rational Number
Imaginary Unit

40. We can also think of the point z= a+ ib as
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z - z*
the vector (a -b)
Liouville's Theorem -

41. Any number not rational
irrational
z1 ^ (z2)
natural
(a + bi) = (c + bi) = (a + c) + ( b + d)i

42. 1
How to solve (2i+3)/(9-i)
Imaginary number
conjugate
cosh²y - sinh²y

43. 2a
z + z*
De Moivre's Theorem
ln z
v(-1)

44. A+bi
Liouville's Theorem -
Absolute Value of a Complex Number
i^0
Complex Number Formula

45. Has exactly n roots by the fundamental theorem of algebra
ln z
subtracting complex numbers
(cos? +isin?)n
Any polynomial O(xn) - (n > 0)

46. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
a real number: (a + bi)(a - bi) = a² + b²
v(-1)
interchangeable

47. Not on the numberline
Subfield
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Conjugate
non-integers

48. I^2 =
-1
Rational Number
i²
Any polynomial O(xn) - (n > 0)

49. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Complex Numbers: Add & subtract
Real and Imaginary Parts
Field
|z-w|

50. 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
e^(ln z)
has a solution.