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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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1. We can also think of the point z= a+ ib as
conjugate
Complex Exponentiation
the vector (a -b)
z1 ^ (z2)

2. When two complex numbers are subtracted from one another.
Complex Subtraction
How to multiply complex nubers(2+i)(2i-3)
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex numbers are points in the plane

3. z1z2* / |z2|²
z1 / z2
Imaginary number
z - z*
x-axis in the complex plane

4. 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
Rules of Complex Arithmetic
Imaginary number
Complex Exponentiation
Real and Imaginary Parts

5. 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.
0 if and only if a = b = 0
i^0
i²
Complex Numbers: Multiply

6. Starts at 1 - does not include 0
z - z*
Complex Exponentiation
natural
|z| = mod(z)

7. (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
0 if and only if a = b = 0
rational
How to solve (2i+3)/(9-i)
zz*

8. 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
adding complex numbers
radicals
i²
Imaginary Unit

9. The modulus of the complex number z= a + ib now can be interpreted as
i²
Complex Subtraction
the distance from z to the origin in the complex plane
Every complex number has the 'Standard Form': a + bi for some real a and b.

10. 1
i^1
i^0
Real and Imaginary Parts
four different numbers: i - -i - 1 - and -1.

11. Numbers on a numberline
has a solution.
Complex Numbers: Add & subtract
integers
four different numbers: i - -i - 1 - and -1.

12. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
cos z
Field
four different numbers: i - -i - 1 - and -1.
Imaginary number

13. When two complex numbers are divided.
Complex Division
the complex numbers
|z| = mod(z)
(a + bi) = (c + bi) = (a + c) + ( b + d)i

14. Like pi
transcendental
i²
Complex Multiplication
a real number: (a + bi)(a - bi) = a² + b²

15. Not on the numberline
zz*
ln z
Polar Coordinates - Division
non-integers

16. I^2 =
Irrational Number
i²
Complex numbers are points in the plane
-1

17. R?¹(cos? - isin?)
i^4
How to solve (2i+3)/(9-i)
Polar Coordinates - z?¹
Polar Coordinates - Division

18. A complex number may be taken to the power of another complex number.
De Moivre's Theorem
Field
Complex Exponentiation
Real Numbers

19. (e^(iz) - e^(-iz)) / 2i
sin z
0 if and only if a = b = 0
cos z
i^3

20. 3rd. Rule of Complex Arithmetic
Complex Exponentiation
For real a and b - a + bi = 0 if and only if a = b = 0
z - z*
rational

21. 2ib
0 if and only if a = b = 0
Imaginary Unit
z - z*
Field

22. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Imaginary Unit
z1 / z2
i^2
conjugate

23. 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
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Numbers
Polar Coordinates - Multiplication

24. ½(e^(iz) + e^(-iz))
cos z
Affix
Polar Coordinates - r
De Moivre's Theorem

25. 1
How to solve (2i+3)/(9-i)
Irrational Number
cosh²y - sinh²y
Complex Number Formula

26. A subset within a field.
v(-1)
De Moivre's Theorem
Complex Addition
Subfield

27. The field of all rational and irrational numbers.
Imaginary number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Real Numbers

28. ½(e^(-y) +e^(y)) = cosh y
cos iy
four different numbers: i - -i - 1 - and -1.
transcendental
The Complex Numbers

29. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
Complex Addition
a + bi for some real a and b.
Euler Formula
subtracting complex numbers

30. Multiply moduli and add arguments
Polar Coordinates - Multiplication
Field
Any polynomial O(xn) - (n > 0)
Roots of Unity

31. R^2 = x
complex
'i'
Imaginary number
Square Root

32. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex numbers are points in the plane
non-integers
Absolute Value of a Complex Number
Complex Numbers: Add & subtract

33. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Argand diagram
Polar Coordinates - cos?
How to find any Power
Roots of Unity

34. To simplify a complex fraction
Complex Number
Complex Conjugate
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex numbers

35. To simplify the square root of a negative number
i^1
Polar Coordinates - cos?
Polar Coordinates - z?¹
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

36. 2nd. Rule of Complex Arithmetic

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

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38. 3
Integers
adding complex numbers
i^3
Complex Number Formula

39. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
complex
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Affix
The Complex Numbers

40. A complex number and its conjugate
Polar Coordinates - Division
i^2 = -1
conjugate pairs
conjugate

41. The product of an imaginary number and its conjugate is
Polar Coordinates - Arg(z*)
For real a and b - a + bi = 0 if and only if a = b = 0
a real number: (a + bi)(a - bi) = a² + b²
x-axis in the complex plane

42. Given (4-2i) the complex conjugate would be (4+2i)
natural
imaginary
De Moivre's Theorem
Complex Conjugate

43. Root negative - has letter i
Complex Addition
i^4
imaginary
standard form of complex numbers

44. All the powers of i can be written as
Irrational Number
four different numbers: i - -i - 1 - and -1.
zz*
z1 ^ (z2)

45. E ^ (z2 ln z1)
z1 ^ (z2)
Every complex number has the 'Standard Form': a + bi for some real a and b.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
complex numbers

46. Any number not rational
-1
'i'
irrational
x-axis in the complex plane

47. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
Complex Addition
Rules of Complex Arithmetic
has a solution.

48. Has exactly n roots by the fundamental theorem of algebra
cosh²y - sinh²y
|z-w|
Any polynomial O(xn) - (n > 0)
rational

49. When two complex numbers are multipiled together.
'i'
For real a and b - a + bi = 0 if and only if a = b = 0
interchangeable
Complex Multiplication

50. 1
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - r
Complex Numbers: Multiply
i^2

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CLEP General Mathematics: Complex Numbers

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