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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 + c) + ( b + d)i
Complex Number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
has a solution.

2. I^2 =
-1
Euler's Formula
'i'
subtracting complex numbers

3. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Any polynomial O(xn) - (n > 0)
The Complex Numbers
Polar Coordinates - z
Roots of Unity

4. A number that can be expressed as a fraction p/q where q is not equal to 0.
Roots of Unity
Rational Number
multiplying complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two

5. A complex number and its conjugate
i^3
Euler's Formula
cos iy
conjugate pairs

6. 2a
z + z*
i^4
'i'
e^(ln z)

7. A subset within a field.
Subfield
Euler Formula
subtracting complex numbers
real

8. 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.
Imaginary Numbers
We say that c+di and c-di are complex conjugates.
How to add and subtract complex numbers (2-3i)-(4+6i)
How to find any Power

9. 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
Rules of Complex Arithmetic
De Moivre's Theorem
Imaginary number

10. The modulus of the complex number z= a + ib now can be interpreted as
Irrational Number
Argand diagram
complex
the distance from z to the origin in the complex plane

11. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
|z| = mod(z)
|z-w|
point of inflection

12. 1
Euler's Formula
i^0
conjugate
Polar Coordinates - Arg(z*)

13. Every complex number has the 'Standard Form':
cosh²y - sinh²y
Complex Multiplication
a + bi for some real a and b.
i^0

14. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Liouville's Theorem -
Argand diagram

15. 1
Complex Subtraction
cosh²y - sinh²y
Imaginary Unit
Complex Number Formula

16. z1z2* / |z2|²
a real number: (a + bi)(a - bi) = a² + b²
complex
z1 / z2
cos iy

17. V(zz*) = v(a² + b²)
|z| = mod(z)
z - z*
The Complex Numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

18. Not on the numberline
z1 / z2
non-integers
Real and Imaginary Parts
For real a and b - a + bi = 0 if and only if a = b = 0

19. The product of an imaginary number and its conjugate is
(cos? +isin?)n
Complex Multiplication
a real number: (a + bi)(a - bi) = a² + b²
transcendental

20. Where the curvature of the graph changes
-1
Square Root
The Complex Numbers
point of inflection

21. 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
imaginary
the complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^1

22. (e^(iz) - e^(-iz)) / 2i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin z
Complex numbers are points in the plane
ln z

23. Have radical
Complex Number
the complex numbers
radicals
subtracting complex numbers

24. V(x² + y²) = |z|
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to solve (2i+3)/(9-i)
subtracting complex numbers
Polar Coordinates - r

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

26. 1
De Moivre's Theorem
i^4
zz*
ln z

27. 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?
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*

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

29. Rotates anticlockwise by p/2
The Complex Numbers
ln z
i^4
Polar Coordinates - Multiplication by i

30. All the powers of i can be written as
Complex numbers are points in the plane
a real number: (a + bi)(a - bi) = a² + b²
x-axis in the complex plane
four different numbers: i - -i - 1 - and -1.

31. R^2 = x
integers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Absolute Value of a Complex Number
Square Root

32. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Euler Formula
Complex Numbers: Add & subtract
Every complex number has the 'Standard Form': a + bi for some real a and b.
Field

33. 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
non-integers
cosh²y - sinh²y
Any polynomial O(xn) - (n > 0)

34. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
z1 / z2
multiplying complex numbers
point of inflection
Any polynomial O(xn) - (n > 0)

35. Divide moduli and subtract arguments
Polar Coordinates - Division
point of inflection
Polar Coordinates - Arg(z*)
Integers

36. 3rd. Rule of Complex Arithmetic
Integers
For real a and b - a + bi = 0 if and only if a = b = 0
Complex numbers are points in the plane
Imaginary Unit

37. 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
Rational Number
Complex Numbers: Add & subtract
Any polynomial O(xn) - (n > 0)
Complex Conjugate

38. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Irrational Number
complex numbers
the distance from z to the origin in the complex plane

39. (e^(-y) - e^(y)) / 2i = i sinh y
Complex Division
z - z*
sin iy
i²

40. 3
conjugate
Liouville's Theorem -
i^3
Complex Multiplication

41. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate pairs
i^2
Complex Addition
conjugate

42. 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
|z-w|
a + bi for some real a and b.
x-axis in the complex plane
adding complex numbers

43. I
Imaginary number
How to add and subtract complex numbers (2-3i)-(4+6i)
four different numbers: i - -i - 1 - and -1.
v(-1)

44. Root negative - has letter i
e^(ln z)
multiplying complex numbers
imaginary
Complex Number Formula

45. Written as fractions - terminating + repeating decimals
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
For real a and b - a + bi = 0 if and only if a = b = 0
rational
i^0

46. (a + bi) = (c + bi) =
Field
e^(ln z)
v(-1)
(a + c) + ( b + d)i

47. Equivalent to an Imaginary Unit.
Field
Argand diagram
Imaginary number
sin z

48. I
i^1
(cos? +isin?)n
can't get out of the complex numbers by adding (or subtracting) or multiplying two
imaginary

49. ½(e^(iz) + e^(-iz))
a real number: (a + bi)(a - bi) = a² + b²
cos z
Roots of Unity
conjugate pairs

50. A + bi
standard form of complex numbers
has a solution.
Complex numbers are points in the plane
Polar Coordinates - z?¹