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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. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
cosh²y - sinh²y
ln z
transcendental
cos iy

2. When two complex numbers are subtracted from one another.
Square Root
Complex Subtraction
z1 / z2
Complex Numbers: Multiply

3. Multiply moduli and add arguments
Polar Coordinates - Multiplication
|z-w|
De Moivre's Theorem
complex

4. Cos n? + i sin n? (for all n integers)
transcendental
i^2 = -1
Polar Coordinates - z?¹
(cos? +isin?)n

5. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
Polar Coordinates - r
Complex Numbers: Add & subtract
'i'

6. In this amazing number field every algebraic equation in z with complex coefficients
x-axis in the complex plane
Subfield
has a solution.
i²

7. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Polar Coordinates - Division
Polar Coordinates - z
The Complex Numbers
cosh²y - sinh²y

8. Like pi
Field
Complex Conjugate
transcendental
a + bi for some real a and b.

9. A+bi
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^0
Complex Number Formula
How to multiply complex nubers(2+i)(2i-3)

10. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
|z| = mod(z)
non-integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

11. R?¹(cos? - isin?)
Polar Coordinates - z?¹
For real a and b - a + bi = 0 if and only if a = b = 0
Integers
z1 / z2

12. 3
i^3
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i²
How to find any Power

13. When two complex numbers are divided.
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Multiplication by i
Complex Division
four different numbers: i - -i - 1 - and -1.

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

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

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16. (a + bi)(c + bi) =
z1 ^ (z2)
How to find any Power
a + bi for some real a and b.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

17. Imaginary number

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18. ? = -tan?
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Arg(z*)
e^(ln z)
Complex numbers are points in the plane

19. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - r
x-axis in the complex plane

20. z1z2* / |z2|²
i^3
z1 / z2
-1
conjugate

21. All the powers of i can be written as
sin z
four different numbers: i - -i - 1 - and -1.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Multiplication

22. The modulus of the complex number z= a + ib now can be interpreted as
cosh²y - sinh²y
i^3
integers
the distance from z to the origin in the complex plane

23. 1st. Rule of Complex Arithmetic
i^2 = -1
i^0
conjugate
Rules of Complex Arithmetic

24. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
'i'
Polar Coordinates - cos?
Polar Coordinates - Arg(z*)

25. 1
i^2 = -1
a real number: (a + bi)(a - bi) = a² + b²
i^0
multiplying complex numbers

26. I
i^3
i^1
Complex Division
ln z

27. A complex number and its conjugate
multiplying complex numbers
conjugate pairs
Polar Coordinates - Arg(z*)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

28. R^2 = x
zz*
'i'
Polar Coordinates - r
Square Root

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
0 if and only if a = b = 0
Subfield
Field
Real and Imaginary Parts

30. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Square Root
i^4
Complex Multiplication

31. A plot of complex numbers as points.
Argand diagram
0 if and only if a = b = 0
The Complex Numbers
How to solve (2i+3)/(9-i)

32. (e^(iz) - e^(-iz)) / 2i
sin z
interchangeable
subtracting complex numbers
complex numbers

33. ½(e^(-y) +e^(y)) = cosh y
|z| = mod(z)
How to find any Power
cos iy
Rules of Complex Arithmetic

34. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the distance from z to the origin in the complex plane
conjugate pairs
a + bi for some real a and b.

35. 2ib
How to add and subtract complex numbers (2-3i)-(4+6i)
multiply the numerator and the denominator by the complex conjugate of the denominator.
z - z*
real

36. 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
Euler's Formula
Absolute Value of a Complex Number
Complex Subtraction
Rules of Complex Arithmetic

37. 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
standard form of complex numbers
adding complex numbers
i^3
four different numbers: i - -i - 1 - and -1.

38. 1
i^2
ln z
Square Root
Complex Conjugate

39. To simplify a complex fraction
adding complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
transcendental
Polar Coordinates - Multiplication by i

40. (a + bi) = (c + bi) =
Polar Coordinates - cos?
Polar Coordinates - r
(a + c) + ( b + d)i
z1 ^ (z2)

41. Derives z = a+bi
i^3
i^0
has a solution.
Euler Formula

42. 1
Euler's Formula
i²
conjugate
(cos? +isin?)n

43. 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
Field
We say that c+di and c-di are complex conjugates.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
can't get out of the complex numbers by adding (or subtracting) or multiplying two

44. Written as fractions - terminating + repeating decimals
|z-w|
rational
z1 / z2
Roots of Unity

45. Not on the numberline
Imaginary Unit
transcendental
non-integers
multiplying complex numbers

46. Every complex number has the 'Standard Form':
Argand diagram
the complex numbers
'i'
a + bi for some real a and b.

47. Starts at 1 - does not include 0
Every complex number has the 'Standard Form': a + bi for some real a and b.
Absolute Value of a Complex Number
natural
z - z*

48. 2a
z + z*
How to multiply complex nubers(2+i)(2i-3)
the distance from z to the origin in the complex plane
Complex numbers are points in the plane

49. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
i^2 = -1
point of inflection
Roots of Unity

50. A subset within a field.
Subfield
Polar Coordinates - sin?
Polar Coordinates - r
rational