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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. V(zz*) = v(a² + b²)
complex
Liouville's Theorem -
can't get out of the complex numbers by adding (or subtracting) or multiplying two
|z| = mod(z)

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

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3. A complex number and its conjugate
Polar Coordinates - z
|z| = mod(z)
conjugate pairs
i^2 = -1

4. Starts at 1 - does not include 0
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Multiplication by i
Complex Number
natural

5. Not on the numberline
'i'
non-integers
the vector (a -b)
complex

6. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
(a + c) + ( b + d)i
-1
Field
i^1

7. Numbers on a numberline
cos iy
integers
Square Root
Complex Exponentiation

8. Have radical
Polar Coordinates - sin?
radicals
complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

9. When two complex numbers are subtracted from one another.
i²
Real and Imaginary Parts
Complex Subtraction
imaginary

10. In this amazing number field every algebraic equation in z with complex coefficients
conjugate
has a solution.
i^4
'i'

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

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12. 1
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^2
Complex numbers are points in the plane
-1

13. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Conjugate
Complex Addition
conjugate
Complex Subtraction

14. E^(ln r) e^(i?) e^(2pin)
z + z*
integers
e^(ln z)
Polar Coordinates - z

15. Cos n? + i sin n? (for all n integers)
i^1
(cos? +isin?)n
Polar Coordinates - Division
radicals

16. (e^(-y) - e^(y)) / 2i = i sinh y
the complex numbers
sin iy
multiply the numerator and the denominator by the complex conjugate of the denominator.
Imaginary Numbers

17. Divide moduli and subtract arguments
z1 / z2
Polar Coordinates - Division
Euler Formula
e^(ln z)

18. 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.
the distance from z to the origin in the complex plane
Complex numbers are points in the plane
(cos? +isin?)n
conjugate pairs

19. I = imaginary unit - i² = -1 or i = v-1
Complex Subtraction
Imaginary Numbers
z1 / z2
i^2 = -1

20. E ^ (z2 ln z1)
Polar Coordinates - Division
Polar Coordinates - Multiplication by i
zz*
z1 ^ (z2)

21. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Rules of Complex Arithmetic
natural
0 if and only if a = b = 0
Absolute Value of a Complex Number

22. R?¹(cos? - isin?)
Complex numbers are points in the plane
Polar Coordinates - z?¹
Imaginary number
z1 ^ (z2)

23. The product of an imaginary number and its conjugate is
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Subfield
a real number: (a + bi)(a - bi) = a² + b²
Complex Addition

24. A + bi
standard form of complex numbers
How to multiply complex nubers(2+i)(2i-3)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)

25. Multiply moduli and add arguments
De Moivre's Theorem
'i'
Polar Coordinates - Multiplication
(cos? +isin?)n

26. ? = -tan?
Complex Numbers: Add & subtract
Polar Coordinates - Arg(z*)
How to multiply complex nubers(2+i)(2i-3)
z - z*

27. Derives z = a+bi
Real and Imaginary Parts
i²
Euler Formula
non-integers

28. Equivalent to an Imaginary Unit.
Imaginary number
Imaginary Unit
How to solve (2i+3)/(9-i)
ln z

29. 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 - Division
irrational
Liouville's Theorem -

30. 5th. Rule of Complex Arithmetic
adding complex numbers
Field
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the distance from z to the origin in the complex plane

31. Like pi
zz*
Square Root
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
transcendental

32. A plot of complex numbers as points.
i^3
Argand diagram
Imaginary Numbers
conjugate

33. 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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34. 1
complex
i^1
i^4
Complex Multiplication

35. x + iy = r(cos? + isin?) = re^(i?)
integers
Complex Subtraction
Polar Coordinates - z
Imaginary Numbers

36. 1
v(-1)
transcendental
cosh²y - sinh²y
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

37. The square root of -1.
the distance from z to the origin in the complex plane
complex
Imaginary Unit
How to solve (2i+3)/(9-i)

38. I
integers
i^1
The Complex Numbers
v(-1)

39. A number that can be expressed as a fraction p/q where q is not equal to 0.
Polar Coordinates - Division
complex numbers
Rational Number
i^3

40. I
z1 ^ (z2)
the vector (a -b)
radicals
v(-1)

41. A number that cannot be expressed as a fraction for any integer.
How to add and subtract complex numbers (2-3i)-(4+6i)
i^4
Irrational Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

42. 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
subtracting complex numbers
Complex Number
complex
Real and Imaginary Parts

43. (a + bi)(c + bi) =
x-axis in the complex plane
multiplying complex numbers
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

44. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - z
Complex Numbers: Add & subtract
Roots of Unity
the complex numbers

45. When two complex numbers are multipiled together.
zz*
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Multiplication
conjugate

46. (e^(iz) - e^(-iz)) / 2i
sin z
z - z*
adding complex numbers
cosh²y - sinh²y

47. Any number not rational
i^2
irrational
Square Root
z1 / z2

48. 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.'
z1 / z2
Complex Number
Euler Formula
transcendental

49. Root negative - has letter i
Subfield
Any polynomial O(xn) - (n > 0)
i^2
imaginary

50. y / r
standard form of complex numbers
Polar Coordinates - sin?
Rules of Complex Arithmetic
the distance from z to the origin in the complex plane

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CLEP General Mathematics: Complex Numbers

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