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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
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. I = imaginary unit - i² = -1 or i = v-1
Polar Coordinates - z
Real Numbers
sin z
Imaginary Numbers

2. V(x² + y²) = |z|
real
Polar Coordinates - r
'i'
Complex Multiplication

3. 1
How to multiply complex nubers(2+i)(2i-3)
cosh²y - sinh²y
sin iy
subtracting complex numbers

4. A + bi
Euler's Formula
Complex Exponentiation
standard form of complex numbers
Complex Division

5. 2nd. Rule of Complex Arithmetic

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6. Cos n? + i sin n? (for all n integers)
i^4
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(cos? +isin?)n
Integers

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

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8. A number that can be expressed as a fraction p/q where q is not equal to 0.
We say that c+di and c-di are complex conjugates.
i^2 = -1
Rational Number
Subfield

9. The product of an imaginary number and its conjugate is
Complex Number Formula
Polar Coordinates - z
a real number: (a + bi)(a - bi) = a² + b²
can't get out of the complex numbers by adding (or subtracting) or multiplying two

10. 2a
z + z*
Polar Coordinates - cos?
Polar Coordinates - z
a real number: (a + bi)(a - bi) = a² + b²

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
subtracting complex numbers
Real and Imaginary Parts
cos z
Polar Coordinates - r

12. (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
Complex Conjugate
cosh²y - sinh²y
Real and Imaginary Parts
How to multiply complex nubers(2+i)(2i-3)

13. All numbers
complex
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Arg(z*)
Complex Number

14. y / r
Polar Coordinates - sin?
Subfield
Polar Coordinates - Multiplication by i
|z| = mod(z)

15. Written as fractions - terminating + repeating decimals
(a + bi) = (c + bi) = (a + c) + ( b + d)i
rational
Euler Formula
|z-w|

16. In this amazing number field every algebraic equation in z with complex coefficients
Integers
natural
conjugate pairs
has a solution.

17. A complex number may be taken to the power of another complex number.
The Complex Numbers
multiplying complex numbers
z1 ^ (z2)
Complex Exponentiation

18. A² + b² - real and non negative
zz*
the vector (a -b)
Imaginary Numbers
Polar Coordinates - Multiplication

19. ? = -tan?
point of inflection
Polar Coordinates - Arg(z*)
Complex Division
How to solve (2i+3)/(9-i)

20. (a + bi) = (c + bi) =
the vector (a -b)
(a + c) + ( b + d)i
z + z*
multiplying complex numbers

21. 3rd. Rule of Complex Arithmetic
Rational Number
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - r
Polar Coordinates - Arg(z*)

22. Have radical
radicals
Roots of Unity
i^2
imaginary

23. Where the curvature of the graph changes
point of inflection
cosh²y - sinh²y
a real number: (a + bi)(a - bi) = a² + b²
the complex numbers

24. A+bi
Complex Number Formula
Complex Conjugate
complex numbers
Complex Number

25. 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
real
Polar Coordinates - z?¹
We say that c+di and c-di are complex conjugates.
Euler Formula

26. Divide moduli and subtract arguments
can't get out of the complex numbers by adding (or subtracting) or multiplying two
zz*
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Division

27. 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.

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28. Imaginary number

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29. z1z2* / |z2|²
Polar Coordinates - Multiplication
a + bi for some real a and b.
i^0
z1 / z2

30. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
natural
conjugate

31. (e^(iz) - e^(-iz)) / 2i
sin z
complex numbers
v(-1)
has a solution.

32. All the powers of i can be written as
i^2
has a solution.
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)

33. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Complex Number
Polar Coordinates - sin?
conjugate

34. Equivalent to an Imaginary Unit.
Imaginary number
Every complex number has the 'Standard Form': a + bi for some real a and b.
subtracting complex numbers
cosh²y - sinh²y

35. 1st. Rule of Complex Arithmetic
(a + c) + ( b + d)i
Complex Addition
Complex Number
i^2 = -1

36. 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
the complex numbers
|z-w|
Complex Multiplication
sin iy

37. Not on the numberline
non-integers
Subfield
Imaginary Numbers
Complex Addition

38. When two complex numbers are subtracted from one another.
x-axis in the complex plane
Complex Subtraction
integers
ln z

39. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
subtracting complex numbers
Square Root
conjugate
i^0

40. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
rational
Imaginary number
Complex numbers are points in the plane

41. Root negative - has letter i
Complex Addition
i^2
Polar Coordinates - sin?
imaginary

42. 1
the distance from z to the origin in the complex plane
i^4
Polar Coordinates - Multiplication
cos iy

43. 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
z1 ^ (z2)
-1
Real and Imaginary Parts
Complex numbers are points in the plane

44. E ^ (z2 ln z1)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z1 ^ (z2)
Real Numbers
|z| = mod(z)

45. 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.
radicals
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Numbers: Multiply

46. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
|z-w|
Complex Number
i^2
adding complex numbers

47. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Complex Conjugate
Polar Coordinates - z
a + bi for some real a and b.

48. 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
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^2 = -1
multiply the numerator and the denominator by the complex conjugate of the denominator.

49. x / r
-1
Polar Coordinates - cos?
Polar Coordinates - z
z1 / z2

50. 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.
subtracting complex numbers
How to find any Power
|z| = mod(z)
Complex Exponentiation