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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. 1
i²
non-integers
x-axis in the complex plane
cos z

2. All numbers
sin z
Imaginary Unit
z + z*
complex

3. xpressions such as ``the complex number z'' - and ``the point z'' are now
How to solve (2i+3)/(9-i)
Polar Coordinates - sin?
interchangeable
Roots of Unity

4. (e^(iz) - e^(-iz)) / 2i
the distance from z to the origin in the complex plane
z1 / z2
zz*
sin z

5. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Polar Coordinates - Multiplication
-1
Polar Coordinates - Division

6. Rotates anticlockwise by p/2
integers
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Multiplication by i
Real and Imaginary Parts

7. The product of an imaginary number and its conjugate is
Polar Coordinates - Division
Complex Conjugate
Roots of Unity
a real number: (a + bi)(a - bi) = a² + b²

8. I
Real and Imaginary Parts
a real number: (a + bi)(a - bi) = a² + b²
i^1
Polar Coordinates - Multiplication by i

9. The field of all rational and irrational numbers.
Real Numbers
(cos? +isin?)n
e^(ln z)
Imaginary number

10. 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: Add & subtract
i²
conjugate
How to multiply complex nubers(2+i)(2i-3)

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

12. 1
Liouville's Theorem -
cosh²y - sinh²y
(a + c) + ( b + d)i
We say that c+di and c-di are complex conjugates.

13. x / r
Polar Coordinates - cos?
i^0
Complex Exponentiation
How to solve (2i+3)/(9-i)

14. Not on the numberline
Complex Conjugate
standard form of complex numbers
the vector (a -b)
non-integers

15. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
v(-1)
sin z
Field
Liouville's Theorem -

16. 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
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - cos?
Complex Division

17. When two complex numbers are divided.
Complex Subtraction
cosh²y - sinh²y
Complex Division
i^4

18. A number that can be expressed as a fraction p/q where q is not equal to 0.
The Complex Numbers
Rational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)

19. Like pi
i^1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
multiply the numerator and the denominator by the complex conjugate of the denominator.
transcendental

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.

21. R^2 = x
Square Root
Any polynomial O(xn) - (n > 0)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Subtraction

22. Derives z = a+bi
sin iy
multiplying complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Euler Formula

23. A plot of complex numbers as points.
Imaginary number
0 if and only if a = b = 0
ln z
Argand diagram

24. 2a
z + z*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^2 = -1
Polar Coordinates - sin?

25. Starts at 1 - does not include 0
Polar Coordinates - cos?
imaginary
For real a and b - a + bi = 0 if and only if a = b = 0
natural

26. A+bi
Euler Formula
Complex Number Formula
z1 / z2
Rational Number

27. Equivalent to an Imaginary Unit.
i^2
Imaginary number
How to find any Power
multiply the numerator and the denominator by the complex conjugate of the denominator.

28. When two complex numbers are multipiled together.
i^2
Complex Multiplication
radicals
(a + c) + ( b + d)i

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

30. Root negative - has letter i
sin iy
multiplying complex numbers
Absolute Value of a Complex Number
imaginary

31. Divide moduli and subtract arguments
Complex Numbers: Multiply
i^2
Polar Coordinates - Division
-1

32. Imaginary number

33. 2nd. Rule of Complex Arithmetic

34. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Affix
How to find any Power
Absolute Value of a Complex Number
x-axis in the complex plane

35. Has exactly n roots by the fundamental theorem of algebra
cosh²y - sinh²y
z - z*
Polar Coordinates - sin?
Any polynomial O(xn) - (n > 0)

36. z1z2* / |z2|²
'i'
(a + c) + ( b + d)i
non-integers
z1 / z2

37. 2ib
e^(ln z)
z - z*
subtracting complex numbers
Complex Exponentiation

38. A subset within a field.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Subfield
Imaginary Numbers
z + z*

39. 1
x-axis in the complex plane
|z-w|
i^0
sin z

40. I
Complex Number Formula
v(-1)
Subfield
Absolute Value of a Complex Number

41. (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
x-axis in the complex plane
How to solve (2i+3)/(9-i)
transcendental

42. Written as fractions - terminating + repeating decimals
subtracting complex numbers
|z-w|
rational
Affix

43. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Square Root
Complex Addition
multiplying complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.

44. The square root of -1.
How to find any Power
Imaginary Unit
Polar Coordinates - Multiplication by i
Polar Coordinates - sin?

45. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
cos z
point of inflection
We say that c+di and c-di are complex conjugates.

46. 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
The Complex Numbers
i^3
Every complex number has the 'Standard Form': a + bi for some real a and b.
We say that c+di and c-di are complex conjugates.

47. 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
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
subtracting complex numbers
Imaginary Numbers
Euler Formula

48. 1st. Rule of Complex Arithmetic
i^2 = -1
Rational Number
i^1
z - z*

49. For real a and b - a + bi =
0 if and only if a = b = 0
has a solution.
Complex Multiplication
subtracting complex numbers

50. 3
i^3
Square Root
z - z*
complex numbers