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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. Any number not rational
irrational
Complex Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to add and subtract complex numbers (2-3i)-(4+6i)

2. 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
0 if and only if a = b = 0
Argand diagram
v(-1)
subtracting complex numbers

3. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Polar Coordinates - Arg(z*)
Rational Number
How to multiply complex nubers(2+i)(2i-3)
Integers

4. We can also think of the point z= a+ ib as
the vector (a -b)
Polar Coordinates - Division
standard form of complex numbers
the complex numbers

5. To simplify the square root of a negative number
(a + c) + ( b + d)i
conjugate pairs
integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

6. Not on the numberline
i^3
non-integers
0 if and only if a = b = 0
For real a and b - a + bi = 0 if and only if a = b = 0

7. 2nd. Rule of Complex Arithmetic

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8. R^2 = x
Absolute Value of a Complex Number
z + z*
Polar Coordinates - Division
Square Root

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

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10. 1
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Subtraction
cosh²y - sinh²y
'i'

11. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
cos z
radicals
Complex numbers are points in the plane

12. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to add and subtract complex numbers (2-3i)-(4+6i)

13. 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
Affix
Polar Coordinates - z?¹
adding complex numbers
Roots of Unity

14. (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
How to solve (2i+3)/(9-i)
a + bi for some real a and b.
Subfield
rational

15. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
irrational
ln z
De Moivre's Theorem
Complex Number

16. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Subfield
Complex Exponentiation
How to find any Power
Roots of Unity

17. 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
i^0
Imaginary Unit
z1 / z2

18. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Imaginary Numbers
interchangeable
real

19. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
four different numbers: i - -i - 1 - and -1.
conjugate
i^1
i²

20. The field of all rational and irrational numbers.
Real Numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
cosh²y - sinh²y
conjugate

21. Imaginary number

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22. A number that cannot be expressed as a fraction for any integer.
How to solve (2i+3)/(9-i)
Irrational Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
adding complex numbers

23. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Liouville's Theorem -
the vector (a -b)
z1 ^ (z2)

24. A subset within a field.
i²
Subfield
(cos? +isin?)n
standard form of complex numbers

25. x / r
Polar Coordinates - cos?
Imaginary Numbers
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Arg(z*)

26. I = imaginary unit - i² = -1 or i = v-1
Polar Coordinates - Multiplication
Imaginary Numbers
i^3
conjugate pairs

27. 3
z1 ^ (z2)
Rational Number
i^3
real

28. V(x² + y²) = |z|
e^(ln z)
Complex Addition
non-integers
Polar Coordinates - r

29. 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
i^0
Polar Coordinates - sin?
multiply the numerator and the denominator by the complex conjugate of the denominator.
We say that c+di and c-di are complex conjugates.

30. To simplify a complex fraction
multiplying complex numbers
adding complex numbers
How to multiply complex nubers(2+i)(2i-3)
multiply the numerator and the denominator by the complex conjugate of the denominator.

31. Root negative - has letter i
Real and Imaginary Parts
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Argand diagram
imaginary

32. Equivalent to an Imaginary Unit.
adding complex numbers
Euler Formula
Imaginary number
subtracting complex numbers

33. 1
adding complex numbers
|z-w|
i^2
i^4

34. No i
the distance from z to the origin in the complex plane
real
adding complex numbers
e^(ln z)

35. 1
i^0
Field
e^(ln z)
real

36. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
multiplying complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i²

37. 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
Square Root
the complex numbers
a real number: (a + bi)(a - bi) = a² + b²
|z| = mod(z)

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

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39. 1
i^0
(a + c) + ( b + d)i
multiplying complex numbers
i^4

40. 1st. Rule of Complex Arithmetic
i^2 = -1
interchangeable
Complex Subtraction
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

41. Has exactly n roots by the fundamental theorem of algebra
Complex Division
has a solution.
Euler Formula
Any polynomial O(xn) - (n > 0)

42. The square root of -1.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Euler Formula
Imaginary Unit
Real Numbers

43. y / r
Polar Coordinates - sin?
irrational
v(-1)
adding complex numbers

44. E ^ (z2 ln z1)
imaginary
i²
Polar Coordinates - Multiplication by i
z1 ^ (z2)

45. 5th. Rule of Complex Arithmetic
Rules of Complex Arithmetic
cos iy
multiplying complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

46. Rotates anticlockwise by p/2
Rational Number
non-integers
Polar Coordinates - Multiplication by i
Complex Exponentiation

47. 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.
Polar Coordinates - sin?
zz*
Euler's Formula
Complex numbers are points in the plane

48. Starts at 1 - does not include 0
natural
How to multiply complex nubers(2+i)(2i-3)
standard form of complex numbers
|z-w|

49. Multiply moduli and add arguments
multiplying complex numbers
Affix
Polar Coordinates - Multiplication
Polar Coordinates - Division

50. 1
the distance from z to the origin in the complex plane
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Division
i²