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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. A+bi
Complex Number Formula
Affix
zz*
i^2

2. All numbers
complex
Complex numbers are points in the plane
The Complex Numbers
How to find any Power

3. Imaginary number

4. Cos n? + i sin n? (for all n integers)
Complex numbers are points in the plane
adding complex numbers
(cos? +isin?)n
Roots of Unity

5. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
Integers
Polar Coordinates - z
|z-w|
How to multiply complex nubers(2+i)(2i-3)

6. 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
cos iy
the distance from z to the origin in the complex plane
Imaginary Numbers
Real and Imaginary Parts

7. Every complex number has the 'Standard Form':
(cos? +isin?)n
a + bi for some real a and b.
Absolute Value of a Complex Number
z - z*

8. A number that can be expressed as a fraction p/q where q is not equal to 0.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rational Number
Any polynomial O(xn) - (n > 0)
ln z

9. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
natural
Euler Formula
z - z*

10. R?¹(cos? - isin?)
Polar Coordinates - z?¹
ln z
x-axis in the complex plane
Complex Number Formula

11. Real and imaginary numbers
How to find any Power
cos z
multiplying complex numbers
complex numbers

12. I
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication
v(-1)
Liouville's Theorem -

13. A number that cannot be expressed as a fraction for any integer.
Irrational Number
'i'
conjugate pairs
Complex Numbers: Add & subtract

14. 1
We say that c+di and c-di are complex conjugates.
non-integers
i^2
i²

15. A subset within a field.
Complex Multiplication
Subfield
(cos? +isin?)n
cosh²y - sinh²y

16. A complex number and its conjugate
Complex Exponentiation
Complex numbers are points in the plane
conjugate pairs
interchangeable

17. x / r
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - cos?
i^2 = -1
z1 / z2

18. E^(ln r) e^(i?) e^(2pin)
0 if and only if a = b = 0
cos z
e^(ln z)
Imaginary Unit

19. Not on the numberline
Polar Coordinates - Multiplication
multiply the numerator and the denominator by the complex conjugate of the denominator.
non-integers
Complex Conjugate

20. 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 numbers are points in the plane
Any polynomial O(xn) - (n > 0)
Complex Number Formula
How to solve (2i+3)/(9-i)

21. To simplify the square root of a negative number
Polar Coordinates - sin?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
conjugate
z1 ^ (z2)

22. Any number not rational
cos z
sin z
Liouville's Theorem -
irrational

23. Derives z = a+bi
Complex Conjugate
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
Euler Formula

24. E ^ (z2 ln z1)
Every complex number has the 'Standard Form': a + bi for some real a and b.
a + bi for some real a and b.
z1 ^ (z2)
i²

25. Starts at 1 - does not include 0
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex numbers are points in the plane
Polar Coordinates - z
natural

26. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Complex Numbers: Add & subtract
Field
Polar Coordinates - z?¹
ln z

27. 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
Imaginary number
Irrational Number
Roots of Unity

28. Where the curvature of the graph changes
transcendental
Affix
i^1
point of inflection

29. The product of an imaginary number and its conjugate is
z + z*
a + bi for some real a and b.
rational
a real number: (a + bi)(a - bi) = a² + b²

30. 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.

31. Has exactly n roots by the fundamental theorem of algebra
Complex Number
How to find any Power
Any polynomial O(xn) - (n > 0)
Complex Subtraction

32. Given (4-2i) the complex conjugate would be (4+2i)
Complex Multiplication
Complex Conjugate
Roots of Unity
multiply the numerator and the denominator by the complex conjugate of the denominator.

33. All the powers of i can be written as
Every complex number has the 'Standard Form': a + bi for some real a and b.
four different numbers: i - -i - 1 - and -1.
conjugate pairs
Affix

34. A complex number may be taken to the power of another complex number.
Polar Coordinates - z
Complex Exponentiation
Every complex number has the 'Standard Form': a + bi for some real a and b.
conjugate pairs

35. 2ib
How to solve (2i+3)/(9-i)
z - z*
Polar Coordinates - z?¹
z1 ^ (z2)

36. 1
adding complex numbers
Polar Coordinates - sin?
Imaginary Numbers
i^4

37. A² + b² - real and non negative
Euler's Formula
How to find any Power
Imaginary number
zz*

38. 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
Irrational Number
integers
Euler's Formula
We say that c+di and c-di are complex conjugates.

39. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
radicals
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
sin iy

40. For real a and b - a + bi =
has a solution.
z - z*
Liouville's Theorem -
0 if and only if a = b = 0

41. When two complex numbers are added together.
the vector (a -b)
Field
Imaginary Unit
Complex Addition

42. (e^(iz) - e^(-iz)) / 2i
Rational Number
Argand diagram
sin z
i^0

43. Divide moduli and subtract arguments
Absolute Value of a Complex Number
e^(ln z)
Polar Coordinates - Division
Complex Multiplication

44. Like pi
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
transcendental
Absolute Value of a Complex Number
non-integers

45. 3
i^3
Complex Number Formula
sin z
(cos? +isin?)n

46. V(x² + y²) = |z|
i^2
Polar Coordinates - r
interchangeable
Rules of Complex Arithmetic

47. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Field
Complex Addition
Roots of Unity
the complex numbers

48. 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.
Polar Coordinates - z
i^0
How to find any Power
Complex Number Formula

49. y / r
Polar Coordinates - sin?
ln z
conjugate pairs
Complex Number Formula

50. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex Division
Absolute Value of a Complex Number
radicals
Imaginary Numbers