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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. 1st. Rule of Complex Arithmetic
Affix
Complex Number Formula
i^2 = -1
zz*

2. The modulus of the complex number z= a + ib now can be interpreted as
Complex Numbers: Multiply
the distance from z to the origin in the complex plane
z1 / z2
Rational Number

3. A number that can be expressed as a fraction p/q where q is not equal to 0.
Argand diagram
Rational Number
z1 / z2
Complex Conjugate

4. Divide moduli and subtract arguments
z1 ^ (z2)
Polar Coordinates - Division
the complex numbers
We say that c+di and c-di are complex conjugates.

5. Given (4-2i) the complex conjugate would be (4+2i)
the vector (a -b)
Complex Conjugate
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Euler's Formula

6. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*
The Complex Numbers
x-axis in the complex plane

7. 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.

8. Have radical
radicals
Field
cos iy
point of inflection

9. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Number Formula
conjugate
De Moivre's Theorem
conjugate pairs

10. y / r
z1 / z2
Polar Coordinates - Multiplication by i
i^4
Polar Coordinates - sin?

11. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
-1
i²
Integers

12. x / r
Polar Coordinates - cos?
z1 ^ (z2)
imaginary
Field

13. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Rational Number
Polar Coordinates - sin?
i^4

14. For real a and b - a + bi =
Complex Numbers: Multiply
(a + bi) = (c + bi) = (a + c) + ( b + d)i
0 if and only if a = b = 0
z - z*

15. Has exactly n roots by the fundamental theorem of algebra
complex numbers
point of inflection
i^3
Any polynomial O(xn) - (n > 0)

16. x + iy = r(cos? + isin?) = re^(i?)
adding complex numbers
i^2
Polar Coordinates - Arg(z*)
Polar Coordinates - z

17. (e^(iz) - e^(-iz)) / 2i
real
sin z
Polar Coordinates - Multiplication
interchangeable

18. xpressions such as ``the complex number z'' - and ``the point z'' are now
subtracting complex numbers
interchangeable
Imaginary Numbers
i²

19. V(x² + y²) = |z|
adding complex numbers
Polar Coordinates - r
i²
How to solve (2i+3)/(9-i)

20. ½(e^(iz) + e^(-iz))
standard form of complex numbers
cos z
conjugate
i^2 = -1

21. 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
Polar Coordinates - Arg(z*)
For real a and b - a + bi = 0 if and only if a = b = 0
We say that c+di and c-di are complex conjugates.
integers

22. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - z?¹
integers
Roots of Unity
imaginary

23. z1z2* / |z2|²
natural
z1 / z2
Complex Exponentiation
Any polynomial O(xn) - (n > 0)

24. I^2 =
Real and Imaginary Parts
How to add and subtract complex numbers (2-3i)-(4+6i)
-1
z - z*

25. 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 are points in the plane
the complex numbers
zz*
real

26. ½(e^(-y) +e^(y)) = cosh y
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos iy
Real Numbers
How to add and subtract complex numbers (2-3i)-(4+6i)

27. Not on the numberline
integers
transcendental
z + z*
non-integers

28. All numbers
Polar Coordinates - Division
can't get out of the complex numbers by adding (or subtracting) or multiplying two
|z-w|
complex

29. E ^ (z2 ln z1)
z1 ^ (z2)
|z-w|
real
We say that c+di and c-di are complex conjugates.

30. Equivalent to an Imaginary Unit.
De Moivre's Theorem
Imaginary number
conjugate pairs
e^(ln z)

31. The product of an imaginary number and its conjugate is
Imaginary number
Square Root
subtracting complex numbers
a real number: (a + bi)(a - bi) = a² + b²

32. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Complex Numbers: Multiply
How to multiply complex nubers(2+i)(2i-3)
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary Unit

33. 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
four different numbers: i - -i - 1 - and -1.
Subfield
Polar Coordinates - r

34. 3rd. Rule of Complex Arithmetic
i^0
Complex Exponentiation
Absolute Value of a Complex Number
For real a and b - a + bi = 0 if and only if a = b = 0

35. Where the curvature of the graph changes
multiplying complex numbers
Polar Coordinates - Multiplication by i
Rules of Complex Arithmetic
point of inflection

36. 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)
radicals
Square Root
We say that c+di and c-di are complex conjugates.

37. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
four different numbers: i - -i - 1 - and -1.
Field
Polar Coordinates - z?¹
rational

38. (a + bi)(c + bi) =
complex
integers
point of inflection
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

39. 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
cosh²y - sinh²y
real
Rules of Complex Arithmetic
Subfield

40. Real and imaginary numbers
complex numbers
adding complex numbers
Polar Coordinates - cos?
a + bi for some real a and b.

41. ? = -tan?
Polar Coordinates - Arg(z*)
Imaginary Numbers
Imaginary number
Complex Numbers: Multiply

42. (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
conjugate pairs
Absolute Value of a Complex Number
(a + c) + ( b + d)i
How to solve (2i+3)/(9-i)

43. 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
Complex Numbers: Multiply
Complex Numbers: Add & subtract
i^2
Polar Coordinates - Multiplication

44. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
rational
four different numbers: i - -i - 1 - and -1.
Irrational Number

45. When two complex numbers are subtracted from one another.
Polar Coordinates - z
Euler's Formula
standard form of complex numbers
Complex Subtraction

46. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
How to find any Power
Complex Numbers: Add & subtract
e^(ln z)
adding complex numbers

47. When two complex numbers are added together.
Complex Addition
Polar Coordinates - sin?
Complex Multiplication
Euler Formula

48. We see in this way that the distance between two points z and w in the complex plane is
The Complex Numbers
Polar Coordinates - r
Argand diagram
|z-w|

49. The field of all rational and irrational numbers.
Complex numbers are points in the plane
Polar Coordinates - Division
Integers
Real Numbers

50. Any number not rational
imaginary
'i'
Field
irrational