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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. Divide moduli and subtract arguments
Polar Coordinates - Division
Integers
Euler's Formula
z - z*

2. R^2 = x
Square Root
Irrational Number
standard form of complex numbers
Integers

3. For real a and b - a + bi =
|z| = mod(z)
0 if and only if a = b = 0
v(-1)
Complex Number Formula

4. When two complex numbers are subtracted from one another.
Euler's Formula
Complex Numbers: Multiply
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Subtraction

5. V(x² + y²) = |z|
Imaginary number
Polar Coordinates - r
cosh²y - sinh²y
Complex Subtraction

6. 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.
Complex numbers are points in the plane
Rules of Complex Arithmetic
the complex numbers
Polar Coordinates - z?¹

7. 1
|z-w|
Field
i^0
Real and Imaginary Parts

8. (e^(-y) - e^(y)) / 2i = i sinh y
multiply the numerator and the denominator by the complex conjugate of the denominator.
-1
sin z
sin iy

9. Has exactly n roots by the fundamental theorem of algebra
the complex numbers
i²
Complex Numbers: Multiply
Any polynomial O(xn) - (n > 0)

10. (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
complex
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)
Euler's Formula

11. 3rd. Rule of Complex Arithmetic
Complex Addition
|z-w|
For real a and b - a + bi = 0 if and only if a = b = 0
the vector (a -b)

12. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Complex numbers are points in the plane
Liouville's Theorem -
the complex numbers

13. I^2 =
Complex Numbers: Multiply
z + z*
-1
Polar Coordinates - sin?

14. Cos n? + i sin n? (for all n integers)
i^2
z + z*
a real number: (a + bi)(a - bi) = a² + b²
(cos? +isin?)n

15. A subset within a field.
point of inflection
Subfield
How to add and subtract complex numbers (2-3i)-(4+6i)
0 if and only if a = b = 0

16. Multiply moduli and add arguments
the distance from z to the origin in the complex plane
adding complex numbers
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication

17. Where the curvature of the graph changes
Complex Number
point of inflection
Polar Coordinates - r
Roots of Unity

18. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

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19. The complex number z representing a+bi.
Affix
complex
Integers
i^2

20. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
subtracting complex numbers
i^0
complex numbers
Absolute Value of a Complex Number

21. ½(e^(iz) + e^(-iz))
Complex Subtraction
The Complex Numbers
cos z
conjugate

22. (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
Real Numbers
natural
How to solve (2i+3)/(9-i)
v(-1)

23. Written as fractions - terminating + repeating decimals
Imaginary Unit
complex
rational
imaginary

24. ? = -tan?
multiplying complex numbers
Complex Numbers: Add & subtract
sin iy
Polar Coordinates - Arg(z*)

25. 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
Real and Imaginary Parts
Complex Numbers: Add & subtract
How to find any Power
the complex numbers

26. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Polar Coordinates - z?¹
Polar Coordinates - z
multiplying complex numbers
Real Numbers

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

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28. 4th. Rule of Complex Arithmetic
ln z
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cosh²y - sinh²y
Irrational Number

29. 1
cosh²y - sinh²y
Rational Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
the complex numbers

30. When two complex numbers are divided.
standard form of complex numbers
non-integers
Complex Division
Square Root

31. E^(ln r) e^(i?) e^(2pin)
Liouville's Theorem -
(a + c) + ( b + d)i
e^(ln z)
z - z*

32. We see in this way that the distance between two points z and w in the complex plane is
adding complex numbers
Complex Division
|z-w|
Polar Coordinates - Multiplication by i

33. Starts at 1 - does not include 0
|z| = mod(z)
Complex Multiplication
a real number: (a + bi)(a - bi) = a² + b²
natural

34. A + bi
Irrational Number
Polar Coordinates - z
Polar Coordinates - r
standard form of complex numbers

35. 2a
conjugate pairs
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z + z*
cosh²y - sinh²y

36. E ^ (z2 ln z1)
(a + c) + ( b + d)i
z1 ^ (z2)
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^3

37. 1st. Rule of Complex Arithmetic
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 ^ (z2)
Integers
i^2 = -1

38. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Subfield
-1
ln z

39. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Rational Number
integers
Field
Polar Coordinates - z

40. 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
cos z
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Multiplication
the distance from z to the origin in the complex plane

41. Real and imaginary numbers
complex numbers
Irrational Number
Polar Coordinates - Multiplication by i
cos iy

42. A+bi
non-integers
Complex Exponentiation
(cos? +isin?)n
Complex Number Formula

43. 1
imaginary
Argand diagram
i²
|z-w|

44. A complex number and its conjugate
Square Root
integers
Polar Coordinates - sin?
conjugate pairs

45. I
i^1
Square Root
How to solve (2i+3)/(9-i)
De Moivre's Theorem

46. 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
Polar Coordinates - sin?
Argand diagram
Rules of Complex Arithmetic
imaginary

47. z1z2* / |z2|²
z1 / z2
Euler's Formula
Complex Number Formula
Liouville's Theorem -

48. 2nd. Rule of Complex Arithmetic

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49. x + iy = r(cos? + isin?) = re^(i?)
transcendental
z1 / z2
Polar Coordinates - z
Complex numbers are points in the plane

50. (e^(iz) - e^(-iz)) / 2i
radicals
sin z
four different numbers: i - -i - 1 - and -1.
e^(ln z)

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CLEP General Mathematics: Complex Numbers

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