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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
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. y / r
Polar Coordinates - sin?
Complex Addition
i^0
irrational

2. A number that cannot be expressed as a fraction for any integer.
subtracting complex numbers
Argand diagram
radicals
Irrational Number

3. I^2 =
four different numbers: i - -i - 1 - and -1.
v(-1)
i^4
-1

4. A+bi
-1
interchangeable
Complex Number Formula
conjugate pairs

5. I
ln z
v(-1)
Real and Imaginary Parts
standard form of complex numbers

6. 1st. Rule of Complex Arithmetic
conjugate pairs
i^2 = -1
v(-1)
Liouville's Theorem -

7. V(zz*) = v(a² + b²)
non-integers
|z| = mod(z)
Polar Coordinates - cos?
adding complex numbers

8. 1
Liouville's Theorem -
sin iy
Polar Coordinates - sin?
i^4

9. R^2 = x
the distance from z to the origin in the complex plane
-1
Complex Numbers: Multiply
Square Root

10. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Complex numbers are points in the plane
Imaginary Numbers
Imaginary number

11. Every complex number has the 'Standard Form':
multiply the numerator and the denominator by the complex conjugate of the denominator.
conjugate pairs
a + bi for some real a and b.
Polar Coordinates - z

12. 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
Complex Addition
i^4
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

13. (e^(iz) - e^(-iz)) / 2i
has a solution.
complex numbers
sin z
v(-1)

14. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Real Numbers
The Complex Numbers
Polar Coordinates - z
multiplying complex numbers

15. Divide moduli and subtract arguments
interchangeable
non-integers
Rational Number
Polar Coordinates - Division

16. The product of an imaginary number and its conjugate is
i^0
a real number: (a + bi)(a - bi) = a² + b²
the distance from z to the origin in the complex plane
four different numbers: i - -i - 1 - and -1.

17. Real and imaginary numbers
Complex Numbers: Multiply
|z| = mod(z)
ln z
complex numbers

18. 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.'
Absolute Value of a Complex Number
Complex Number
Imaginary number
ln z

19. 1
Complex Exponentiation
Complex Division
Complex Number
i^2

20. Equivalent to an Imaginary Unit.
four different numbers: i - -i - 1 - and -1.
multiplying complex numbers
the vector (a -b)
Imaginary number

21. 1
cos z
Subfield
i^0
Absolute Value of a Complex Number

22. A complex number may be taken to the power of another complex number.
has a solution.
Complex Exponentiation
Complex Addition
0 if and only if a = b = 0

23. 2nd. Rule of Complex Arithmetic

24. ½(e^(-y) +e^(y)) = cosh y
We say that c+di and c-di are complex conjugates.
cos iy
How to multiply complex nubers(2+i)(2i-3)
i^4

25. A subset within a field.
Complex Numbers: Multiply
Complex Multiplication
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Subfield

26. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
the distance from z to the origin in the complex plane
point of inflection
four different numbers: i - -i - 1 - and -1.

27. R?¹(cos? - isin?)
has a solution.
Polar Coordinates - z?¹
i²
Irrational Number

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

29. We can also think of the point z= a+ ib as
'i'
|z| = mod(z)
Polar Coordinates - Multiplication by i
the vector (a -b)

30. Where the curvature of the graph changes
Polar Coordinates - z?¹
0 if and only if a = b = 0
point of inflection
Complex Multiplication

31. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
De Moivre's Theorem
a + bi for some real a and b.
subtracting complex numbers

32. Numbers on a numberline
integers
We say that c+di and c-di are complex conjugates.
'i'
Rules of Complex Arithmetic

33. A complex number and its conjugate
Complex Addition
the complex numbers
(cos? +isin?)n
conjugate pairs

34. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
How to find any Power
sin iy
Polar Coordinates - z?¹
conjugate

35. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
Field
x-axis in the complex plane
Complex Addition

36. Written as fractions - terminating + repeating decimals
Complex Numbers: Multiply
rational
Complex Addition
Every complex number has the 'Standard Form': a + bi for some real a and b.

37. A plot of complex numbers as points.
Polar Coordinates - Multiplication by i
cos z
Argand diagram
i²

38. 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
interchangeable
Complex Addition
i^0
adding complex numbers

39. 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
i^1
Liouville's Theorem -
Rules of Complex Arithmetic
Square Root

40. 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 Exponentiation
Liouville's Theorem -
Real Numbers
Complex Numbers: Add & subtract

41. Have radical
Integers
radicals
has a solution.
sin z

42. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Complex Multiplication
Real Numbers
imaginary

43. 4th. Rule of Complex Arithmetic
Polar Coordinates - Multiplication by i
|z-w|
point of inflection
(a + bi) = (c + bi) = (a + c) + ( b + d)i

44. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
the complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
four different numbers: i - -i - 1 - and -1.

45. Derives z = a+bi
Euler Formula
standard form of complex numbers
Irrational Number
Complex Addition

46. Like pi
i^0
|z| = mod(z)
transcendental
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

47. 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 - Arg(z*)
cos z
multiplying complex numbers
subtracting complex numbers

48. I = imaginary unit - i² = -1 or i = v-1
Euler's Formula
z - z*
x-axis in the complex plane
Imaginary Numbers

49. A number that can be expressed as a fraction p/q where q is not equal to 0.
sin iy
sin z
Rational Number
Complex Number

50. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
z + z*
Complex numbers are points in the plane
i^4
For real a and b - a + bi = 0 if and only if a = b = 0