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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.
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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. When two complex numbers are multipiled together.
|z-w|
Complex Multiplication
How to solve (2i+3)/(9-i)
Polar Coordinates - z?¹

2. Divide moduli and subtract arguments
interchangeable
Complex numbers are points in the plane
Polar Coordinates - Division
the vector (a -b)

3. 2ib
z - z*
Complex Numbers: Multiply
|z-w|
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

4. 5th. Rule of Complex Arithmetic
i²
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

5. Written as fractions - terminating + repeating decimals
i^3
rational
How to multiply complex nubers(2+i)(2i-3)
Imaginary Numbers

6. 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
conjugate pairs
Any polynomial O(xn) - (n > 0)
the complex numbers
How to find any Power

7. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
real
subtracting complex numbers
Rational Number

8. Have radical
sin z
has a solution.
Irrational Number
radicals

9. A number that can be expressed as a fraction p/q where q is not equal to 0.
Irrational Number
Rational Number
(a + c) + ( b + d)i
Affix

10. When two complex numbers are subtracted from one another.
Complex Subtraction
irrational
Complex Conjugate
i²

11. In this amazing number field every algebraic equation in z with complex coefficients
Imaginary number
has a solution.
Polar Coordinates - sin?
Polar Coordinates - z

12. ? = -tan?
Subfield
real
Euler Formula
Polar Coordinates - Arg(z*)

13. 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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14. 3
Polar Coordinates - Division
i^3
conjugate pairs
How to solve (2i+3)/(9-i)

15. y / r
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
Polar Coordinates - sin?
How to solve (2i+3)/(9-i)

16. Any number not rational
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Argand diagram
irrational
Real Numbers

17. A+bi
Complex Number Formula
'i'
z - z*
Complex Division

18. z1z2* / |z2|²
z1 / z2
z - z*
sin iy
Polar Coordinates - Multiplication

19. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Integers
Field
ln z
v(-1)

20. 1
Imaginary number
Polar Coordinates - sin?
Affix
i^0

21. I
Liouville's Theorem -
v(-1)
cos iy
Rules of Complex Arithmetic

22. For real a and b - a + bi =
Polar Coordinates - Division
Complex Multiplication
0 if and only if a = b = 0
i^4

23. Derives z = a+bi
Complex Exponentiation
We say that c+di and c-di are complex conjugates.
Affix
Euler Formula

24. 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
cos z
i^1
adding complex numbers
Rational Number

25. x / r
the distance from z to the origin in the complex plane
Polar Coordinates - cos?
Euler Formula
Complex Subtraction

26. 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.'
Complex Number
'i'
For real a and b - a + bi = 0 if and only if a = b = 0
cosh²y - sinh²y

27. 2nd. Rule of Complex Arithmetic

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28. I^2 =
Complex numbers are points in the plane
point of inflection
natural
-1

29. We can also think of the point z= a+ ib as
four different numbers: i - -i - 1 - and -1.
the vector (a -b)
zz*
Complex Conjugate

30. (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 - Multiplication by i
How to multiply complex nubers(2+i)(2i-3)
-1
|z| = mod(z)

31. 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 - sin?
How to find any Power
Complex Exponentiation

32. 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
multiply the numerator and the denominator by the complex conjugate of the denominator.
subtracting complex numbers
How to solve (2i+3)/(9-i)
radicals

33. 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 and Imaginary Parts
Complex Exponentiation
Euler's Formula
complex

34. Not on the numberline
has a solution.
0 if and only if a = b = 0
non-integers
Polar Coordinates - Multiplication by i

35. 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^1
Square Root
point of inflection

36. 1
i^2
standard form of complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
How to find any Power

37. Like pi
the vector (a -b)
Polar Coordinates - cos?
z + z*
transcendental

38. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
i²
Field
non-integers
How to solve (2i+3)/(9-i)

39. Every complex number has the 'Standard Form':
a + bi for some real a and b.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
e^(ln z)
Argand diagram

40. The reals are just the
Real Numbers
x-axis in the complex plane
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex numbers are points in the plane

41. A subset within a field.
conjugate
complex
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Subfield

42. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
How to find any Power
We say that c+di and c-di are complex conjugates.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

43. A complex number may be taken to the power of another complex number.
The Complex Numbers
ln z
Complex Exponentiation
Real and Imaginary Parts

44. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Subfield
point of inflection
Integers
|z| = mod(z)

45. 1
Complex Conjugate
i^4
Absolute Value of a Complex Number
real

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

47. The product of an imaginary number and its conjugate is
Complex Numbers: Add & subtract
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - cos?
x-axis in the complex plane

48. When two complex numbers are added together.
Polar Coordinates - z?¹
Complex Addition
irrational
'i'

49. 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.
e^(ln z)
How to find any Power
Complex numbers are points in the plane
Complex Exponentiation

50. Equivalent to an Imaginary Unit.
Polar Coordinates - Multiplication
|z| = mod(z)
Imaginary number
Complex Number Formula