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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. Real and imaginary numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary Numbers
complex numbers
Polar Coordinates - Division

2. Numbers on a numberline
the vector (a -b)
integers
i^0
can't get out of the complex numbers by adding (or subtracting) or multiplying two

3. V(zz*) = v(a² + b²)
|z| = mod(z)
i^4
interchangeable
irrational

4. 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.
Argand diagram
Complex Numbers: Add & subtract
i^2 = -1
How to find any Power

5. x / r
Rational Number
Polar Coordinates - cos?
(cos? +isin?)n
radicals

6. z1z2* / |z2|²
z1 / z2
Polar Coordinates - Division
How to add and subtract complex numbers (2-3i)-(4+6i)
(cos? +isin?)n

7. To simplify the square root of a negative number
i^0
i^2 = -1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
|z| = mod(z)

8. 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.
Complex Numbers: Multiply
Polar Coordinates - Multiplication by i
rational
Imaginary Numbers

9. 1
Absolute Value of a Complex Number
non-integers
radicals
cosh²y - sinh²y

10. I = imaginary unit - i² = -1 or i = v-1
zz*
i^0
Imaginary Numbers
How to solve (2i+3)/(9-i)

11. A complex number may be taken to the power of another complex number.
Complex Exponentiation
z1 / z2
Rules of Complex Arithmetic
zz*

12. 3
rational
i^3
sin z
transcendental

13. 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
complex numbers
i^1
the complex numbers
Imaginary Numbers

14. ½(e^(iz) + e^(-iz))
adding complex numbers
cos z
imaginary
Subfield

15. R?¹(cos? - isin?)
Polar Coordinates - z?¹
non-integers
|z-w|
Imaginary Numbers

16. In this amazing number field every algebraic equation in z with complex coefficients
conjugate
has a solution.
Liouville's Theorem -
multiplying complex numbers

17. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
point of inflection
Polar Coordinates - z?¹
Complex Exponentiation

18. x + iy = r(cos? + isin?) = re^(i?)
Complex Numbers: Multiply
Real and Imaginary Parts
Real Numbers
Polar Coordinates - z

19. E ^ (z2 ln z1)
irrational
z1 ^ (z2)
the distance from z to the origin in the complex plane
Field

20. 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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21. We can also think of the point z= a+ ib as
cos z
the vector (a -b)
z - z*
non-integers

22. No i
cosh²y - sinh²y
Imaginary number
real
rational

23. Like pi
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
transcendental
standard form of complex numbers
How to find any Power

24. Has exactly n roots by the fundamental theorem of algebra
Imaginary number
real
Any polynomial O(xn) - (n > 0)
Complex Addition

25. For real a and b - a + bi =
conjugate pairs
0 if and only if a = b = 0
Polar Coordinates - r
|z-w|

26. Cos n? + i sin n? (for all n integers)
Irrational Number
(cos? +isin?)n
radicals
i^0

27. 1
real
radicals
Complex Addition
i^2

28. 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.
complex
Complex numbers are points in the plane
(a + c) + ( b + d)i
i^4

29. 1st. Rule of Complex Arithmetic
i^2 = -1
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Conjugate
conjugate

30. 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
Imaginary number
Rules of Complex Arithmetic
Polar Coordinates - Multiplication by i
Irrational Number

31. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
i^2 = -1
Real and Imaginary Parts
(cos? +isin?)n
Imaginary Unit

32. The product of an imaginary number and its conjugate is
the distance from z to the origin in the complex plane
a real number: (a + bi)(a - bi) = a² + b²
Real Numbers
Complex Exponentiation

33. E^(ln r) e^(i?) e^(2pin)
four different numbers: i - -i - 1 - and -1.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
point of inflection
e^(ln z)

34. (a + bi) = (c + bi) =
Irrational Number
Complex Numbers: Add & subtract
0 if and only if a = b = 0
(a + c) + ( b + d)i

35. Not on the numberline
sin z
Complex Number
Imaginary number
non-integers

36. A + bi
Complex Multiplication
standard form of complex numbers
Complex Numbers: Add & subtract
Imaginary Numbers

37. Root negative - has letter i
i^4
imaginary
Affix
Imaginary Unit

38. Where the curvature of the graph changes
transcendental
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
point of inflection
i^4

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

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40. When two complex numbers are added together.
Complex Addition
Affix
multiplying complex numbers
Polar Coordinates - z

41. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
Complex Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos iy
We say that c+di and c-di are complex conjugates.

42. 1
cos z
i^0
standard form of complex numbers
natural

43. xpressions such as ``the complex number z'' - and ``the point z'' are now
Affix
Polar Coordinates - sin?
Complex Numbers: Add & subtract
interchangeable

44. To simplify a complex fraction
z - z*
Absolute Value of a Complex Number
Complex Exponentiation
multiply the numerator and the denominator by the complex conjugate of the denominator.

45. We see in this way that the distance between two points z and w in the complex plane is
Polar Coordinates - Multiplication by i
The Complex Numbers
Polar Coordinates - cos?
|z-w|

46. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Real Numbers
Complex Exponentiation
Euler's Formula
How to add and subtract complex numbers (2-3i)-(4+6i)

47. When two complex numbers are subtracted from one another.
radicals
Complex Subtraction
i^1
interchangeable

48. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
multiplying complex numbers
conjugate
(a + c) + ( b + d)i
ln z

49. (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
v(-1)
How to solve (2i+3)/(9-i)
z - z*
sin iy

50. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary Numbers
a + bi for some real a and b.
adding complex numbers