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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. 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
a + bi for some real a and b.
point of inflection
Any polynomial O(xn) - (n > 0)

2. A subset within a field.
Subfield
Polar Coordinates - cos?
multiplying complex numbers
Polar Coordinates - Division

3. When two complex numbers are subtracted from one another.
Polar Coordinates - Arg(z*)
Complex Subtraction
zz*
imaginary

4. Where the curvature of the graph changes
Any polynomial O(xn) - (n > 0)
Polar Coordinates - cos?
a real number: (a + bi)(a - bi) = a² + b²
point of inflection

5. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17
Imaginary number
How to solve (2i+3)/(9-i)
Complex Addition
|z| = mod(z)

6. Rotates anticlockwise by p/2
i^3
The Complex Numbers
Polar Coordinates - Multiplication by i
conjugate

7. The field of all rational and irrational numbers.
Real Numbers
How to find any Power
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)

8. 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
How to multiply complex nubers(2+i)(2i-3)
Any polynomial O(xn) - (n > 0)
Polar Coordinates - z
Rules of Complex Arithmetic

9. 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
Euler Formula
the vector (a -b)
Argand diagram
Real and Imaginary Parts

10. 2ib
z - z*
Roots of Unity
cos z
Complex Exponentiation

11. All numbers
Complex Numbers: Multiply
four different numbers: i - -i - 1 - and -1.
complex
conjugate pairs

12. Equivalent to an Imaginary Unit.
Any polynomial O(xn) - (n > 0)
radicals
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary number

13. Multiply moduli and add arguments
De Moivre's Theorem
We say that c+di and c-di are complex conjugates.
z1 / z2
Polar Coordinates - Multiplication

14. I
How to solve (2i+3)/(9-i)
v(-1)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
has a solution.

15. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Complex Subtraction
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

16. We can also think of the point z= a+ ib as
the vector (a -b)
cos iy
How to solve (2i+3)/(9-i)
real

17. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
i^1
Subfield
|z| = mod(z)

18. 5th. Rule of Complex Arithmetic
e^(ln z)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Subtraction
Complex Numbers: Add & subtract

19. R^2 = x
Square Root
Every complex number has the 'Standard Form': a + bi for some real a and b.
i²
Complex Numbers: Add & subtract

20. 1st. Rule of Complex Arithmetic
imaginary
interchangeable
Complex Subtraction
i^2 = -1

21. 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
z1 ^ (z2)
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Subtraction
We say that c+di and c-di are complex conjugates.

22. (a + bi)(c + bi) =
x-axis in the complex plane
Field
Polar Coordinates - Multiplication by i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

23. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Roots of Unity
How to add and subtract complex numbers (2-3i)-(4+6i)
cosh²y - sinh²y

24. A+bi
Complex Number Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.
the distance from z to the origin in the complex plane
Polar Coordinates - Multiplication by i

25. 1
Field
a real number: (a + bi)(a - bi) = a² + b²
the vector (a -b)
i^0

26. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Roots of Unity
ln z
Argand diagram

27. V(x² + y²) = |z|
interchangeable
Any polynomial O(xn) - (n > 0)
Complex Multiplication
Polar Coordinates - r

28. Real and imaginary numbers
complex numbers
Imaginary Numbers
non-integers
multiplying complex numbers

29. (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
How to multiply complex nubers(2+i)(2i-3)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Multiplication
Irrational Number

30. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Complex Addition
irrational
|z| = mod(z)

31. Given (4-2i) the complex conjugate would be (4+2i)
a real number: (a + bi)(a - bi) = a² + b²
Complex Conjugate
sin iy
standard form of complex numbers

32. When two complex numbers are divided.
Polar Coordinates - r
Complex Division
natural
irrational

33. ? = -tan?
z + z*
(cos? +isin?)n
Polar Coordinates - Arg(z*)
Polar Coordinates - cos?

34. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
non-integers
z - z*
How to multiply complex nubers(2+i)(2i-3)
ln z

35. 4th. Rule of Complex Arithmetic
a real number: (a + bi)(a - bi) = a² + b²
How to solve (2i+3)/(9-i)
Absolute Value of a Complex Number
(a + bi) = (c + bi) = (a + c) + ( b + d)i

36. 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 Addition
Complex Numbers: Add & subtract
Euler's Formula
i^3

37. 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.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Numbers: Multiply
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos z

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

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39. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Conjugate
Liouville's Theorem -
cos z
conjugate

40. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
has a solution.
four different numbers: i - -i - 1 - and -1.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

41. When two complex numbers are added together.
Argand diagram
multiplying complex numbers
Imaginary number
Complex Addition

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
Square Root
i^2 = -1
adding complex numbers
irrational

43. E^(ln r) e^(i?) e^(2pin)
cos iy
non-integers
e^(ln z)
rational

44. Written as fractions - terminating + repeating decimals
a real number: (a + bi)(a - bi) = a² + b²
Complex Number Formula
rational
cosh²y - sinh²y

45. A number that can be expressed as a fraction p/q where q is not equal to 0.
Complex Conjugate
Polar Coordinates - z
Affix
Rational Number

46. Like pi
De Moivre's Theorem
irrational
i^4
transcendental

47. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
How to multiply complex nubers(2+i)(2i-3)
e^(ln z)
Polar Coordinates - Division

48. z1z2* / |z2|²
zz*
Complex Conjugate
can't get out of the complex numbers by adding (or subtracting) or multiplying two
z1 / z2

49. 1
Liouville's Theorem -
Polar Coordinates - Division
Complex Exponentiation
i²

50. Numbers on a numberline
the vector (a -b)
integers
i^2
(a + c) + ( b + d)i