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CLEP General Mathematics: Complex Numbers

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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. R?¹(cos? - isin?)
Polar Coordinates - z?¹
The Complex Numbers
i^2
i^2 = -1

2. ? = -tan?
standard form of complex numbers
|z-w|
Complex Number
Polar Coordinates - Arg(z*)

3. Has exactly n roots by the fundamental theorem of algebra
adding complex numbers
Any polynomial O(xn) - (n > 0)
For real a and b - a + bi = 0 if and only if a = b = 0
Complex numbers are points in the plane

4. The complex number z representing a+bi.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Affix
Field
Integers

5. 2a
Integers
z + z*
imaginary
i²

6. Imaginary number

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7. A + bi
the complex numbers
z1 ^ (z2)
Polar Coordinates - cos?
standard form of complex numbers

8. Not on the numberline
non-integers
Subfield
the vector (a -b)
Complex Addition

9. Any number not rational
Complex Numbers: Multiply
Square Root
irrational
z + z*

10. A number that can be expressed as a fraction p/q where q is not equal to 0.
zz*
Rational Number
complex
Roots of Unity

11. (e^(iz) - e^(-iz)) / 2i
sin z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^2
z1 / z2

12. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
z1 ^ (z2)
Integers
Polar Coordinates - z?¹

13. 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
adding complex numbers
Polar Coordinates - Division
Imaginary number

14. ½(e^(-y) +e^(y)) = cosh y
multiply the numerator and the denominator by the complex conjugate of the denominator.
cos iy
Real and Imaginary Parts
Polar Coordinates - cos?

15. (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
Polar Coordinates - r
How to multiply complex nubers(2+i)(2i-3)
conjugate pairs
i^2

16. 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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17. I
i^1
real
i^0
x-axis in the complex plane

18. The product of an imaginary number and its conjugate is
Complex Numbers: Add & subtract
|z-w|
real
a real number: (a + bi)(a - bi) = a² + b²

19. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
interchangeable
adding complex numbers
cosh²y - sinh²y

20. A complex number and its conjugate
Complex Number
conjugate pairs
Argand diagram
zz*

21. When two complex numbers are subtracted from one another.
x-axis in the complex plane
Complex Subtraction
e^(ln z)
conjugate pairs

22. Starts at 1 - does not include 0
Polar Coordinates - Division
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
natural
i²

23. We see in this way that the distance between two points z and w in the complex plane is
Rational Number
|z-w|
Liouville's Theorem -
sin iy

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

25. 1
|z| = mod(z)
Polar Coordinates - cos?
cosh²y - sinh²y
i^2

26. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
i^1
|z| = mod(z)
Complex Multiplication
Integers

27. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

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28. For real a and b - a + bi =
(a + c) + ( b + d)i
0 if and only if a = b = 0
Complex Numbers: Add & subtract
sin z

29. (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
'i'
How to solve (2i+3)/(9-i)
Field
Complex Number Formula

30. 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.'
Complex Number
z + z*
Roots of Unity
Complex Numbers: Add & subtract

31. (e^(-y) - e^(y)) / 2i = i sinh y
has a solution.
sin iy
Real and Imaginary Parts
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

32. 2nd. Rule of Complex Arithmetic

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33. x + iy = r(cos? + isin?) = re^(i?)
How to multiply complex nubers(2+i)(2i-3)
Complex numbers are points in the plane
We say that c+di and c-di are complex conjugates.
Polar Coordinates - z

34. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
-1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex numbers are points in the plane

35. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
i^2
Polar Coordinates - Multiplication
Absolute Value of a Complex Number
Euler's Formula

36. I
Square Root
Complex Numbers: Multiply
Polar Coordinates - sin?
v(-1)

37. A² + b² - real and non negative
(a + c) + ( b + d)i
complex
Affix
zz*

38. 3
Complex Multiplication
Rules of Complex Arithmetic
Complex Numbers: Multiply
i^3

39. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
i^0
Rules of Complex Arithmetic
i^4

40. I = imaginary unit - i² = -1 or i = v-1
Irrational Number
Imaginary Numbers
the complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

41. 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
Polar Coordinates - Division
How to find any Power
adding complex numbers
Complex Number Formula

42. Where the curvature of the graph changes
point of inflection
has a solution.
Complex Conjugate
conjugate pairs

43. The square root of -1.
Imaginary Unit
'i'
Euler's Formula
Imaginary Numbers

44. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Exponentiation
cos iy
Absolute Value of a Complex Number

45. A plot of complex numbers as points.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a + bi for some real a and b.
conjugate pairs
Argand diagram

46. The reals are just the
x-axis in the complex plane
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Euler Formula
Every complex number has the 'Standard Form': a + bi for some real a and b.

47. All numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex
Field
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

48. The field of all rational and irrational numbers.
Real Numbers
For real a and b - a + bi = 0 if and only if a = b = 0
the distance from z to the origin in the complex plane
Field

49. 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
Complex Conjugate
Real and Imaginary Parts
rational
Polar Coordinates - Division

50. 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
the complex numbers
multiplying complex numbers
Polar Coordinates - Division
multiply the numerator and the denominator by the complex conjugate of the denominator.

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CLEP General Mathematics: Complex Numbers

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