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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. Equivalent to an Imaginary Unit.
zz*
Imaginary number
ln z
the vector (a -b)

2. Numbers on a numberline
cosh²y - sinh²y
integers
Argand diagram
Irrational Number

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

4. Where the curvature of the graph changes
The Complex Numbers
0 if and only if a = b = 0
Polar Coordinates - Arg(z*)
point of inflection

5. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
i²
the complex numbers
Integers
How to add and subtract complex numbers (2-3i)-(4+6i)

6. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

7. V(x² + y²) = |z|
The Complex Numbers
'i'
Polar Coordinates - r
Absolute Value of a Complex Number

8. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
the distance from z to the origin in the complex plane
complex
Absolute Value of a Complex Number
cos iy

9. A complex number may be taken to the power of another complex number.
four different numbers: i - -i - 1 - and -1.
Real Numbers
standard form of complex numbers
Complex Exponentiation

10. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Addition
Any polynomial O(xn) - (n > 0)
multiplying complex numbers

11. 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
a real number: (a + bi)(a - bi) = a² + b²
subtracting complex numbers
z - z*
standard form of complex numbers

12. Rotates anticlockwise by p/2
integers
Complex Division
v(-1)
Polar Coordinates - Multiplication by i

13. 1
i^2
Real Numbers
complex numbers
rational

14. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
e^(ln z)
Polar Coordinates - z
i^2

15. When two complex numbers are subtracted from one another.
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - z?¹
Complex Subtraction
multiplying complex numbers

16. Root negative - has letter i
the vector (a -b)
Imaginary Unit
imaginary
Integers

17. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
i²
Imaginary Numbers
complex
conjugate

18. V(zz*) = v(a² + b²)
conjugate
cosh²y - sinh²y
cos z
|z| = mod(z)

19. I
z1 ^ (z2)
i^1
irrational
Polar Coordinates - Arg(z*)

20. A² + b² - real and non negative
e^(ln z)
For real a and b - a + bi = 0 if and only if a = b = 0
i^4
zz*

21. The reals are just the
conjugate
Imaginary Unit
subtracting complex numbers
x-axis in the complex plane

22. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
has a solution.
Subfield
sin iy

23. Like pi
multiplying complex numbers
z - z*
transcendental
z1 / z2

24. A + bi
Polar Coordinates - r
standard form of complex numbers
conjugate
(a + bi) = (c + bi) = (a + c) + ( b + d)i

25. When two complex numbers are divided.
|z| = mod(z)
cosh²y - sinh²y
Complex Division
complex

26. When two complex numbers are added together.
Polar Coordinates - Multiplication by i
Polar Coordinates - sin?
integers
Complex Addition

27. The square root of -1.
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Unit
Every complex number has the 'Standard Form': a + bi for some real a and b.
cos iy

28. For real a and b - a + bi =
0 if and only if a = b = 0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
point of inflection
We say that c+di and c-di are complex conjugates.

29. 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
transcendental
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
has a solution.

30. y / r
How to multiply complex nubers(2+i)(2i-3)
imaginary
Polar Coordinates - sin?
Every complex number has the 'Standard Form': a + bi for some real a and b.

31. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
transcendental
Polar Coordinates - Multiplication
De Moivre's Theorem

32. 2a
|z| = mod(z)
conjugate pairs
z + z*
Polar Coordinates - Multiplication by i

33. x / r
integers
Liouville's Theorem -
Polar Coordinates - cos?
Polar Coordinates - Multiplication by i

34. 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.

35. 1
Polar Coordinates - Division
How to solve (2i+3)/(9-i)
Square Root
i²

36. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - sin?
Complex Addition
i^3

37. 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
ln z
cosh²y - sinh²y
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

38. To simplify the square root of a negative number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
|z-w|
Polar Coordinates - Arg(z*)

39. 2nd. Rule of Complex Arithmetic

40. To simplify a complex fraction
Real and Imaginary Parts
Square Root
multiply the numerator and the denominator by the complex conjugate of the denominator.
can't get out of the complex numbers by adding (or subtracting) or multiplying two

41. Starts at 1 - does not include 0
point of inflection
We say that c+di and c-di are complex conjugates.
z1 / z2
natural

42. Divide moduli and subtract arguments
Polar Coordinates - Division
has a solution.
0 if and only if a = b = 0
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

43. Real and imaginary numbers
complex numbers
Roots of Unity
Subfield
-1

44. x + iy = r(cos? + isin?) = re^(i?)
Field
Polar Coordinates - z
complex numbers
Real Numbers

45. xpressions such as ``the complex number z'' - and ``the point z'' are now
Any polynomial O(xn) - (n > 0)
interchangeable
a + bi for some real a and b.
Complex Number

46. Not on the numberline
Complex Conjugate
cosh²y - sinh²y
x-axis in the complex plane
non-integers

47. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
i^1
Irrational Number
0 if and only if a = b = 0
Field

48. We can also think of the point z= a+ ib as
the vector (a -b)
Any polynomial O(xn) - (n > 0)
i^1
Complex Conjugate

49. 4th. Rule of Complex Arithmetic
Complex Number
the distance from z to the origin in the complex plane
Complex Exponentiation
(a + bi) = (c + bi) = (a + c) + ( b + d)i

50. Have radical
i^2 = -1
Complex Multiplication
How to multiply complex nubers(2+i)(2i-3)
radicals