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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
Field
Complex Number Formula
Irrational Number
the complex numbers

2. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
multiplying complex numbers
standard form of complex numbers
We say that c+di and c-di are complex conjugates.

3. Written as fractions - terminating + repeating decimals
rational
e^(ln z)
Polar Coordinates - Multiplication by i
Square Root

4. 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.
Imaginary number
How to find any Power
i^2 = -1
Complex Addition

5. 1st. Rule of Complex Arithmetic
i^2 = -1
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to multiply complex nubers(2+i)(2i-3)
complex

6. When two complex numbers are subtracted from one another.
sin iy
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Conjugate
Complex Subtraction

7. 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.
i^2
the distance from z to the origin in the complex plane
conjugate pairs
Complex numbers are points in the plane

8. (a + bi) = (c + bi) =
Complex Subtraction
interchangeable
(a + c) + ( b + d)i
Polar Coordinates - sin?

9. Rotates anticlockwise by p/2
transcendental
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication by i
Complex Conjugate

10. y / r
Polar Coordinates - Multiplication
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - sin?
real

11. A number that cannot be expressed as a fraction for any integer.
Irrational Number
interchangeable
Argand diagram
complex

12. 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
Rules of Complex Arithmetic
adding complex numbers
Polar Coordinates - Division
Polar Coordinates - r

13. We can also think of the point z= a+ ib as
the vector (a -b)
cos z
z1 / z2
Complex Exponentiation

14. V(x² + y²) = |z|
Irrational Number
Subfield
Euler Formula
Polar Coordinates - r

15. Have radical
Polar Coordinates - r
radicals
the distance from z to the origin in the complex plane
Complex Conjugate

16. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Polar Coordinates - Arg(z*)
i^2 = -1
conjugate

17. A subset within a field.
Subfield
i^2
Polar Coordinates - Multiplication
v(-1)

18. 3
Polar Coordinates - cos?
Affix
complex
i^3

19. The square root of -1.
radicals
Imaginary Unit
i^2
cos z

20. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
|z| = mod(z)
The Complex Numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Absolute Value of a Complex Number

21. R?¹(cos? - isin?)
Polar Coordinates - z?¹
Integers
sin iy
conjugate pairs

22. Every complex number has the 'Standard Form':
a + bi for some real a and b.
Imaginary Unit
v(-1)
Polar Coordinates - Multiplication

23. (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
i^0
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Number
How to solve (2i+3)/(9-i)

24. Where the curvature of the graph changes
point of inflection
Absolute Value of a Complex Number
imaginary
cos iy

25. Divide moduli and subtract arguments
Polar Coordinates - Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Liouville's Theorem -
Rules of Complex Arithmetic

26. Equivalent to an Imaginary Unit.
Imaginary number
How to find any Power
cosh²y - sinh²y
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

27. Not on the numberline
Imaginary Numbers
Complex Exponentiation
radicals
non-integers

28. 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
i^4
the complex numbers
ln z
transcendental

29. 2ib
the distance from z to the origin in the complex plane
z - z*
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Multiplication by i

30. xpressions such as ``the complex number z'' - and ``the point z'' are now
|z| = mod(z)
Complex numbers are points in the plane
For real a and b - a + bi = 0 if and only if a = b = 0
interchangeable

31. I
adding complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^1

32. 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
We say that c+di and c-di are complex conjugates.
Polar Coordinates - cos?
How to multiply complex nubers(2+i)(2i-3)
conjugate

33. 2a
z + z*
Complex Addition
Liouville's Theorem -
Imaginary number

34. Root negative - has letter i
cosh²y - sinh²y
imaginary
Every complex number has the 'Standard Form': a + bi for some real a and b.
-1

35. 1
imaginary
(a + bi) = (c + bi) = (a + c) + ( b + d)i
v(-1)
i^0

36. Given (4-2i) the complex conjugate would be (4+2i)
Polar Coordinates - r
Complex Conjugate
Euler's Formula
cosh²y - sinh²y

37. 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.
adding complex numbers
Complex Numbers: Multiply
i^2 = -1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

38. Starts at 1 - does not include 0
natural
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number
Polar Coordinates - Multiplication by i

39. 4th. Rule of Complex Arithmetic
i^0
conjugate
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi) = (c + bi) = (a + c) + ( b + d)i

40. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - sin?
Roots of Unity
z - z*

41. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
the distance from z to the origin in the complex plane
Integers
(a + c) + ( b + d)i
How to add and subtract complex numbers (2-3i)-(4+6i)

42. z1z2* / |z2|²
z1 / z2
four different numbers: i - -i - 1 - and -1.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
complex numbers

43. ? = -tan?
0 if and only if a = b = 0
Polar Coordinates - Arg(z*)
Polar Coordinates - sin?
How to solve (2i+3)/(9-i)

44. 1
imaginary
the vector (a -b)
i^0
i^2

45. E ^ (z2 ln z1)
x-axis in the complex plane
Real Numbers
z1 ^ (z2)
Roots of Unity

46. I = imaginary unit - i² = -1 or i = v-1
Every complex number has the 'Standard Form': a + bi for some real a and b.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Add & subtract
Imaginary Numbers

47. V(zz*) = v(a² + b²)
For real a and b - a + bi = 0 if and only if a = b = 0
adding complex numbers
|z| = mod(z)
We say that c+di and c-di are complex conjugates.

48. The field of all rational and irrational numbers.
i^0
Real Numbers
Complex Division
Rational Number

49. The reals are just the
transcendental
x-axis in the complex plane
Square Root
Euler's Formula

50. (e^(iz) - e^(-iz)) / 2i
sin z
interchangeable
x-axis in the complex plane
Complex Multiplication