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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. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Polar Coordinates - z
cosh²y - sinh²y
Absolute Value of a Complex Number
(a + c) + ( b + d)i

2. z1z2* / |z2|²
Complex Conjugate
z1 / z2
Absolute Value of a Complex Number
natural

3. Imaginary number

4. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
'i'
Imaginary number
Complex Multiplication
Integers

5. (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
Polar Coordinates - z?¹
Polar Coordinates - Division
Imaginary number
How to multiply complex nubers(2+i)(2i-3)

6. A+bi
Roots of Unity
z1 ^ (z2)
Complex Number Formula
cos z

7. Has exactly n roots by the fundamental theorem of algebra
v(-1)
i^1
Any polynomial O(xn) - (n > 0)
i^0

8. A subset within a field.
Complex Exponentiation
Subfield
multiplying complex numbers
z - z*

9. xpressions such as ``the complex number z'' - and ``the point z'' are now
conjugate
z1 / z2
interchangeable
Complex Addition

10. To simplify the square root of a negative number
How to add and subtract complex numbers (2-3i)-(4+6i)
a + bi for some real a and b.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
De Moivre's Theorem

11. All the powers of i can be written as
Subfield
(a + bi) = (c + bi) = (a + c) + ( b + d)i
e^(ln z)
four different numbers: i - -i - 1 - and -1.

12. ½(e^(iz) + e^(-iz))
cos z
Field
Rules of Complex Arithmetic
Polar Coordinates - z?¹

13. 1
cos z
i²
i^3
'i'

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

15. 2a
How to find any Power
z + z*
i^2 = -1
i^3

16. When two complex numbers are divided.
Polar Coordinates - sin?
Any polynomial O(xn) - (n > 0)
Complex Numbers: Add & subtract
Complex Division

17. I
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^1
'i'
cosh²y - sinh²y

18. 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
Polar Coordinates - r
Complex Numbers: Add & subtract
Complex numbers are points in the plane
z + z*

19. I = imaginary unit - i² = -1 or i = v-1
natural
a + bi for some real a and b.
Imaginary Numbers
v(-1)

20. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - r
How to add and subtract complex numbers (2-3i)-(4+6i)
Euler's Formula

21. We see in this way that the distance between two points z and w in the complex plane is
sin z
|z-w|
v(-1)
Rules of Complex Arithmetic

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

23. A complex number and its conjugate
transcendental
How to solve (2i+3)/(9-i)
Square Root
conjugate pairs

24. The complex number z representing a+bi.
x-axis in the complex plane
i²
We say that c+di and c-di are complex conjugates.
Affix

25. A number that cannot be expressed as a fraction for any integer.
subtracting complex numbers
(a + c) + ( b + d)i
Euler Formula
Irrational Number

26. Real and imaginary numbers
imaginary
complex numbers
e^(ln z)
irrational

27. The field of all rational and irrational numbers.
point of inflection
The Complex Numbers
imaginary
Real Numbers

28. x / r
How to find any Power
Polar Coordinates - cos?
-1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

29. 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
Real Numbers
Real and Imaginary Parts
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary number

30. 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
irrational
Square Root
the complex numbers
Euler Formula

31. y / r
How to solve (2i+3)/(9-i)
Polar Coordinates - sin?
i^0
|z| = mod(z)

32. 3
i^3
Affix
z1 / z2
multiply the numerator and the denominator by the complex conjugate of the denominator.

33. For real a and b - a + bi =
Roots of Unity
De Moivre's Theorem
adding complex numbers
0 if and only if a = b = 0

34. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
point of inflection
adding complex numbers
e^(ln z)
How to add and subtract complex numbers (2-3i)-(4+6i)

35. V(x² + y²) = |z|
zz*
non-integers
Polar Coordinates - r
Complex Multiplication

36. 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.'
Real Numbers
Polar Coordinates - z
x-axis in the complex plane
Complex Number

37. Have radical
Complex Division
radicals
cosh²y - sinh²y
Complex numbers are points in the plane

38. 2ib
Polar Coordinates - Arg(z*)
z - z*
complex
point of inflection

39. Numbers on a numberline
Every complex number has the 'Standard Form': a + bi for some real a and b.
integers
cos iy
the vector (a -b)

40. When two complex numbers are subtracted from one another.
the complex numbers
real
Polar Coordinates - cos?
Complex Subtraction

41. 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.
a + bi for some real a and b.
standard form of complex numbers
Complex Numbers: Multiply
De Moivre's Theorem

42. R^2 = x
Square Root
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Argand diagram
conjugate

43. A plot of complex numbers as points.
Argand diagram
z1 / z2
interchangeable
a real number: (a + bi)(a - bi) = a² + b²

44. Starts at 1 - does not include 0
the vector (a -b)
integers
We say that c+di and c-di are complex conjugates.
natural

45. Multiply moduli and add arguments
four different numbers: i - -i - 1 - and -1.
Complex Exponentiation
Polar Coordinates - Multiplication
Rules of Complex Arithmetic

46. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Subfield
complex
Complex Addition

47. Root negative - has letter i
x-axis in the complex plane
imaginary
multiply the numerator and the denominator by the complex conjugate of the denominator.
adding complex numbers

48. 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
Complex Numbers: Add & subtract
subtracting complex numbers
x-axis in the complex plane
z1 ^ (z2)

49. 5th. Rule of Complex Arithmetic
subtracting complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number
i^3

50. 1. i^2 = -1 2. Every complex number has the 'Standard Form': a + bi for some real a and b. 3. For real a and b - a + bi = 0 if and only if a = b = 0 4. (a + bi) = (c + bi) = (a + c) + ( b + d)i 5. (a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc
z1 / z2
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rules of Complex Arithmetic
multiplying complex numbers