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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.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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 this amazing number field every algebraic equation in z with complex coefficients
e^(ln z)
Affix
has a solution.
Imaginary Numbers

2. 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.
Absolute Value of a Complex Number
How to find any Power
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
zz*

3. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex Number
conjugate
Absolute Value of a Complex Number
Affix

4. A+bi
Complex Number Formula
Integers
0 if and only if a = b = 0
Complex Multiplication

5. (a + bi) = (c + bi) =
How to solve (2i+3)/(9-i)
Complex Exponentiation
(a + c) + ( b + d)i
Imaginary Numbers

6. Written as fractions - terminating + repeating decimals
rational
point of inflection
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Absolute Value of a Complex Number

7. For real a and b - a + bi =
How to add and subtract complex numbers (2-3i)-(4+6i)
multiply the numerator and the denominator by the complex conjugate of the denominator.
0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

8. (a + bi)(c + bi) =
(cos? +isin?)n
Irrational Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

9. All the powers of i can be written as
sin iy
Polar Coordinates - z
imaginary
four different numbers: i - -i - 1 - and -1.

10. Every complex number has the 'Standard Form':
Complex Division
can't get out of the complex numbers by adding (or subtracting) or multiplying two
a + bi for some real a and b.
standard form of complex numbers

11. x + iy = r(cos? + isin?) = re^(i?)
|z| = mod(z)
Polar Coordinates - Multiplication
Polar Coordinates - z
non-integers

12. I
0 if and only if a = b = 0
Square Root
i^1
adding complex numbers

13. 1
z1 / z2
i^4
the complex numbers
Polar Coordinates - sin?

14. ½(e^(-y) +e^(y)) = cosh y
z1 / z2
complex numbers
Complex Conjugate
cos iy

15. R?¹(cos? - isin?)
Polar Coordinates - z?¹
x-axis in the complex plane
Any polynomial O(xn) - (n > 0)
Field

16. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Integers
Subfield
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

17. (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
can't get out of the complex numbers by adding (or subtracting) or multiplying two
cosh²y - sinh²y
Irrational Number
How to solve (2i+3)/(9-i)

18. 1
i^2
(a + c) + ( b + d)i
subtracting complex numbers
Liouville's Theorem -

19. I^2 =
i^3
-1
Polar Coordinates - Multiplication
a real number: (a + bi)(a - bi) = a² + b²

20. I
v(-1)
zz*
imaginary
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

21. 2ib
z - z*
irrational
Polar Coordinates - z?¹
0 if and only if a = b = 0

22. 1
-1
i^0
Imaginary number
real

23. Multiply moduli and add arguments
Complex numbers are points in the plane
Polar Coordinates - Multiplication
i^2
Complex Numbers: Add & subtract

24. Where the curvature of the graph changes
Rules of Complex Arithmetic
has a solution.
How to multiply complex nubers(2+i)(2i-3)
point of inflection

25. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
cosh²y - sinh²y
real
How to add and subtract complex numbers (2-3i)-(4+6i)
i^3

26. I = imaginary unit - i² = -1 or i = v-1
i²
How to solve (2i+3)/(9-i)
Imaginary Numbers
Complex Number Formula

27. A number that cannot be expressed as a fraction for any integer.
Irrational Number
For real a and b - a + bi = 0 if and only if a = b = 0
point of inflection
How to solve (2i+3)/(9-i)

28. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
v(-1)
Integers
a + bi for some real a and b.
sin iy

29. 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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30. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
multiplying complex numbers
Absolute Value of a Complex Number
ln z
i^2 = -1

31. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
z1 ^ (z2)
The Complex Numbers
Complex Addition
i^4

32. E ^ (z2 ln z1)
z1 ^ (z2)
v(-1)
Rules of Complex Arithmetic
Euler's Formula

33. When two complex numbers are subtracted from one another.
Polar Coordinates - sin?
Complex Exponentiation
Complex Subtraction
can't get out of the complex numbers by adding (or subtracting) or multiplying two

34. 1
cosh²y - sinh²y
(a + c) + ( b + d)i
Rational Number
Complex Conjugate

35. A number that can be expressed as a fraction p/q where q is not equal to 0.
Complex Subtraction
Euler Formula
Every complex number has the 'Standard Form': a + bi for some real a and b.
Rational Number

36. 3rd. Rule of Complex Arithmetic
(cos? +isin?)n
irrational
For real a and b - a + bi = 0 if and only if a = b = 0
How to add and subtract complex numbers (2-3i)-(4+6i)

37. V(zz*) = v(a² + b²)
complex
|z| = mod(z)
v(-1)
imaginary

38. Imaginary number

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39. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Polar Coordinates - sin?
Polar Coordinates - Multiplication by i
How to multiply complex nubers(2+i)(2i-3)

40. Have radical
ln z
radicals
Polar Coordinates - sin?
z1 ^ (z2)

41. x / r
Polar Coordinates - cos?
non-integers
Complex Multiplication
Real Numbers

42. No i
Complex Subtraction
Polar Coordinates - r
Polar Coordinates - z
real

43. 3
i^3
Imaginary Unit
cosh²y - sinh²y
Field

44. A + bi
standard form of complex numbers
v(-1)
real
Field

45. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
i^0
-1
a + bi for some real a and b.

46. Divide moduli and subtract arguments
multiplying complex numbers
i^2
Polar Coordinates - Division
conjugate pairs

47. When two complex numbers are divided.
Complex Division
ln z
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - cos?

48. Starts at 1 - does not include 0
Polar Coordinates - Multiplication
(cos? +isin?)n
natural
Polar Coordinates - z

49. When two complex numbers are added together.
How to solve (2i+3)/(9-i)
Complex Addition
Complex Numbers: Multiply
i^0

50. Rotates anticlockwise by p/2
Polar Coordinates - r
Polar Coordinates - Multiplication by i
Complex Subtraction
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i