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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. A number that cannot be expressed as a fraction for any integer.
a real number: (a + bi)(a - bi) = a² + b²
Irrational Number
natural
multiplying complex numbers

2. (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
Roots of Unity
has a solution.
i^2
How to multiply complex nubers(2+i)(2i-3)

3. x / r
irrational
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - cos?
z + z*

4. We see in this way that the distance between two points z and w in the complex plane is
Polar Coordinates - z
Complex Exponentiation
|z-w|
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

5. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
(cos? +isin?)n
i^4
Roots of Unity

6. 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
rational
We say that c+di and c-di are complex conjugates.
How to solve (2i+3)/(9-i)
Complex numbers are points in the plane

7. Written as fractions - terminating + repeating decimals
the vector (a -b)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
radicals
rational

8. Derives z = a+bi
Complex Division
Euler Formula
Imaginary Unit
v(-1)

9. 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.
ln z
Complex Numbers: Multiply
a + bi for some real a and b.
How to multiply complex nubers(2+i)(2i-3)

10. I
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)
We say that c+di and c-di are complex conjugates.
Argand diagram

11. A plot of complex numbers as points.
the complex numbers
How to solve (2i+3)/(9-i)
Argand diagram
|z| = mod(z)

12. I = imaginary unit - i² = -1 or i = v-1
Complex Numbers: Multiply
Imaginary Numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
sin z

13. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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14. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
Complex Number Formula
irrational
Imaginary number

15. 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
Polar Coordinates - Multiplication by i
subtracting complex numbers
Affix
Complex Number

16. 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.'
i^1
Polar Coordinates - Multiplication by i
Complex Number
Square Root

17. R?¹(cos? - isin?)
Imaginary Unit
Roots of Unity
a + bi for some real a and b.
Polar Coordinates - z?¹

18. The field of all rational and irrational numbers.
Polar Coordinates - z?¹
The Complex Numbers
Real Numbers
(a + c) + ( b + d)i

19. R^2 = x
complex
Complex Conjugate
Square Root
Euler Formula

20. ½(e^(-y) +e^(y)) = cosh y
cos iy
i^1
Rules of Complex Arithmetic
z - z*

21. V(x² + y²) = |z|
i^0
Real Numbers
Polar Coordinates - r
We say that c+di and c-di are complex conjugates.

22. The modulus of the complex number z= a + ib now can be interpreted as
has a solution.
v(-1)
the distance from z to the origin in the complex plane
rational

23. Like pi
De Moivre's Theorem
the vector (a -b)
transcendental
Liouville's Theorem -

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

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25. I^2 =
How to solve (2i+3)/(9-i)
-1
interchangeable
cosh²y - sinh²y

26. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Subfield
conjugate
adding complex numbers
v(-1)

27. Cos n? + i sin n? (for all n integers)
Liouville's Theorem -
Every complex number has the 'Standard Form': a + bi for some real a and b.
(cos? +isin?)n
v(-1)

28. No i
Polar Coordinates - cos?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
real
Field

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
(a + c) + ( b + d)i
Polar Coordinates - Division
imaginary
the complex numbers

30. 4th. Rule of Complex Arithmetic
Roots of Unity
the distance from z to the origin in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - z

31. 1
Subfield
cosh²y - sinh²y
How to add and subtract complex numbers (2-3i)-(4+6i)
For real a and b - a + bi = 0 if and only if a = b = 0

32. When two complex numbers are subtracted from one another.
Any polynomial O(xn) - (n > 0)
i²
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Subtraction

33. 1
imaginary
i^0
transcendental
can't get out of the complex numbers by adding (or subtracting) or multiplying two

34. x + iy = r(cos? + isin?) = re^(i?)
cos iy
conjugate
Polar Coordinates - z
Euler's Formula

35. Numbers on a numberline
z + z*
standard form of complex numbers
integers
Complex Division

36. To simplify a complex fraction
Euler Formula
i^3
imaginary
multiply the numerator and the denominator by the complex conjugate of the denominator.

37. I
i^1
the distance from z to the origin in the complex plane
ln z
a real number: (a + bi)(a - bi) = a² + b²

38. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

39. z1z2* / |z2|²
cos z
Polar Coordinates - z
z1 / z2
Euler Formula

40. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
Euler Formula
Polar Coordinates - Arg(z*)
complex numbers

41. E ^ (z2 ln z1)
z1 ^ (z2)
conjugate
standard form of complex numbers
i^2 = -1

42. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Every complex number has the 'Standard Form': a + bi for some real a and b.
(a + c) + ( b + d)i
imaginary

43. 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
conjugate
adding complex numbers
Absolute Value of a Complex Number
Square Root

44. 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
Complex Multiplication
We say that c+di and c-di are complex conjugates.
Rules of Complex Arithmetic
x-axis in the complex plane

45. The reals are just the
z1 / z2
x-axis in the complex plane
Euler Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

46. Imaginary number

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47. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
How to solve (2i+3)/(9-i)
z1 / z2
conjugate
Field

48. Every complex number has the 'Standard Form':
a + bi for some real a and b.
|z| = mod(z)
i^1
|z-w|

49. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Real and Imaginary Parts
Polar Coordinates - z?¹
ln z
Euler Formula

50. For real a and b - a + bi =
0 if and only if a = b = 0
Imaginary Unit
ln z
Complex Number