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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. 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.'
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number
Irrational Number
the distance from z to the origin in the complex plane

2. For real a and b - a + bi =
|z-w|
How to multiply complex nubers(2+i)(2i-3)
z + z*
0 if and only if a = b = 0

3. V(zz*) = v(a² + b²)
|z| = mod(z)
Roots of Unity
z1 ^ (z2)
complex numbers

4. To simplify the square root of a negative number
conjugate pairs
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
has a solution.
ln z

5. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
subtracting complex numbers
standard form of complex numbers

6. Imaginary number

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7. 3
i^3
the distance from z to the origin in the complex plane
We say that c+di and c-di are complex conjugates.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

8. Real and imaginary numbers
Rational Number
Integers
sin z
complex numbers

9. Any number not rational
(a + c) + ( b + d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division
irrational

10. (a + bi) = (c + bi) =
|z| = mod(z)
(a + c) + ( b + d)i
(cos? +isin?)n
How to solve (2i+3)/(9-i)

11. 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
We say that c+di and c-di are complex conjugates.
Rules of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
multiplying complex numbers

12. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Addition
i^3
multiplying complex numbers
Complex Division

13. ½(e^(-y) +e^(y)) = cosh y
Any polynomial O(xn) - (n > 0)
cos iy
Complex Numbers: Multiply
Irrational Number

14. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
z1 ^ (z2)
Any polynomial O(xn) - (n > 0)
How to solve (2i+3)/(9-i)

15. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
i^1
(cos? +isin?)n
|z| = mod(z)

16. 5th. Rule of Complex Arithmetic
can't get out of the complex numbers by adding (or subtracting) or multiplying two
integers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

17. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Integers
Complex Subtraction
De Moivre's Theorem

18. Derives z = a+bi
Euler Formula
i^3
Polar Coordinates - Arg(z*)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

19. The product of an imaginary number and its conjugate is
For real a and b - a + bi = 0 if and only if a = b = 0
a real number: (a + bi)(a - bi) = a² + b²
cos z
multiply the numerator and the denominator by the complex conjugate of the denominator.

20. A + bi
standard form of complex numbers
z + z*
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - z

21. 1
i²
Complex numbers are points in the plane
four different numbers: i - -i - 1 - and -1.
Complex Subtraction

22. ? = -tan?
Argand diagram
How to find any Power
conjugate
Polar Coordinates - Arg(z*)

23. E ^ (z2 ln z1)
z1 ^ (z2)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - z?¹

24. Divide moduli and subtract arguments
natural
Polar Coordinates - z?¹
Polar Coordinates - Division
The Complex Numbers

25. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
How to find any Power
Complex Number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Integers

26. Have radical
Rules of Complex Arithmetic
z + z*
radicals
Field

27. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
For real a and b - a + bi = 0 if and only if a = b = 0
conjugate pairs
a real number: (a + bi)(a - bi) = a² + b²

28. A number that cannot be expressed as a fraction for any integer.
Irrational Number
ln z
Polar Coordinates - Arg(z*)
sin iy

29. 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
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - cos?
i^2 = -1
the complex numbers

30. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
the complex numbers
How to solve (2i+3)/(9-i)
Every complex number has the 'Standard Form': a + bi for some real a and b.

31. I = imaginary unit - i² = -1 or i = v-1
Polar Coordinates - r
x-axis in the complex plane
sin iy
Imaginary Numbers

32. Multiply moduli and add arguments
i^2
-1
Polar Coordinates - Multiplication
Euler Formula

33. We can also think of the point z= a+ ib as
the vector (a -b)
i²
ln z
Imaginary Numbers

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

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35. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Division
conjugate
transcendental
(a + bi) = (c + bi) = (a + c) + ( b + d)i

36. (a + bi)(c + bi) =
ln z
Polar Coordinates - r
0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

37. 2ib
How to find any Power
z - z*
the complex numbers
Any polynomial O(xn) - (n > 0)

38. 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.
Polar Coordinates - cos?
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Rational Number

39. All the powers of i can be written as
zz*
real
natural
four different numbers: i - -i - 1 - and -1.

40. 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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41. When two complex numbers are subtracted from one another.
Field
non-integers
has a solution.
Complex Subtraction

42. We see in this way that the distance between two points z and w in the complex plane is
Polar Coordinates - Multiplication
|z-w|
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)

43. (e^(-y) - e^(y)) / 2i = i sinh y
irrational
sin iy
sin z
Irrational Number

44. (e^(iz) - e^(-iz)) / 2i
How to multiply complex nubers(2+i)(2i-3)
Affix
sin z
four different numbers: i - -i - 1 - and -1.

45. When two complex numbers are added together.
complex
Complex Addition
Polar Coordinates - Multiplication
Imaginary Numbers

46. Numbers on a numberline
Complex Number
Roots of Unity
integers
i^2

47. 2nd. Rule of Complex Arithmetic

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48. I^2 =
0 if and only if a = b = 0
cos iy
-1
z1 ^ (z2)

49. 1
i^0
cosh²y - sinh²y
Complex Number
the complex numbers

50. x / r
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - cos?
Square Root
Complex Numbers: Multiply