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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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1. (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 numbers
How to multiply complex nubers(2+i)(2i-3)
Rules of Complex Arithmetic
Complex Numbers: Multiply

2. The square root of -1.
Complex Subtraction
Complex Addition
Any polynomial O(xn) - (n > 0)
Imaginary Unit

3. Real and imaginary numbers
(a + c) + ( b + d)i
complex numbers
|z-w|
Complex Addition

4. Rotates anticlockwise by p/2
ln z
Polar Coordinates - Multiplication by i
Polar Coordinates - Arg(z*)
real

5. 3
i^3
Absolute Value of a Complex Number
z1 / z2
Complex Exponentiation

6. 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
subtracting complex numbers
-1
Complex Division
Field

7. x / r
the distance from z to the origin in the complex plane
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - cos?
zz*

8. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
z1 / z2
Real Numbers
How to solve (2i+3)/(9-i)
How to add and subtract complex numbers (2-3i)-(4+6i)

9. Multiply moduli and add arguments
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
four different numbers: i - -i - 1 - and -1.
zz*
Polar Coordinates - Multiplication

10. 1
i^4
Complex Exponentiation
Liouville's Theorem -
the complex numbers

11. The reals are just the
x-axis in the complex plane
Integers
Polar Coordinates - Multiplication by i
De Moivre's Theorem

12. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
x-axis in the complex plane
For real a and b - a + bi = 0 if and only if a = b = 0
How to find any Power
sin iy

13. R^2 = x
i²
Square Root
Liouville's Theorem -
imaginary

14. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Imaginary Unit
irrational
ln z
Complex Numbers: Multiply

15. For real a and b - a + bi =
rational
Imaginary Numbers
-1
0 if and only if a = b = 0

16. R?¹(cos? - isin?)
Polar Coordinates - z?¹
Euler's Formula
Irrational Number
Complex Subtraction

17. V(zz*) = v(a² + b²)
non-integers
|z| = mod(z)
Imaginary Numbers
Irrational Number

18. All the powers of i can be written as
z1 ^ (z2)
natural
Polar Coordinates - z?¹
four different numbers: i - -i - 1 - and -1.

19. Starts at 1 - does not include 0
(cos? +isin?)n
natural
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to find any Power

20. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
the complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Multiplication
Field

21. A number that cannot be expressed as a fraction for any integer.
|z-w|
Irrational Number
conjugate
Complex numbers are points in the plane

22. Not on the numberline
non-integers
standard form of complex numbers
Polar Coordinates - sin?
sin iy

23. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Imaginary Unit
point of inflection
Integers
Complex Exponentiation

24. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
imaginary
Affix
Roots of Unity
conjugate

25. To simplify the square root of a negative number
point of inflection
(cos? +isin?)n
Polar Coordinates - z
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

26. (a + bi)(c + bi) =
Complex Number Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex numbers
point of inflection

27. 1
How to find any Power
How to add and subtract complex numbers (2-3i)-(4+6i)
i²
a + bi for some real a and b.

28. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
i^4
ln z
-1

29. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
Rational Number
Complex numbers are points in the plane
i^0

30. Given (4-2i) the complex conjugate would be (4+2i)
Imaginary Unit
Complex Conjugate
non-integers
z1 ^ (z2)

31. When two complex numbers are divided.
Complex Subtraction
Complex Division
i^2
a real number: (a + bi)(a - bi) = a² + b²

32. I
i^1
(a + c) + ( b + d)i
Roots of Unity
irrational

33. When two complex numbers are multipiled together.
multiplying complex numbers
Complex Multiplication
i^2 = -1
z + z*

34. A+bi
Complex Number Formula
Polar Coordinates - Multiplication
Polar Coordinates - Division
How to multiply complex nubers(2+i)(2i-3)

35. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
zz*
irrational
Complex numbers are points in the plane

36. ½(e^(iz) + e^(-iz))
Polar Coordinates - z
transcendental
cos z
'i'

37. 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
i²
subtracting complex numbers
i^1
Rules of Complex Arithmetic

38. We can also think of the point z= a+ ib as
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the vector (a -b)
x-axis in the complex plane
real

39. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
a real number: (a + bi)(a - bi) = a² + b²
x-axis in the complex plane
Irrational Number

40. Has exactly n roots by the fundamental theorem of algebra
Complex Numbers: Multiply
Complex Multiplication
integers
Any polynomial O(xn) - (n > 0)

41. 1
Polar Coordinates - Multiplication by i
integers
natural
i^0

42. Written as fractions - terminating + repeating decimals
(a + bi) = (c + bi) = (a + c) + ( b + d)i
For real a and b - a + bi = 0 if and only if a = b = 0
integers
rational

43. I
(a + c) + ( b + d)i
v(-1)
has a solution.
Real and Imaginary Parts

44. To simplify a complex fraction
Real and Imaginary Parts
integers
complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.

45. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
De Moivre's Theorem
Polar Coordinates - sin?
Field
multiplying complex numbers

46. 1
v(-1)
i^2
Complex Exponentiation
How to find any Power

47. 1st. Rule of Complex Arithmetic
(a + c) + ( b + d)i
conjugate pairs
Complex Division
i^2 = -1

48. Derives z = a+bi
natural
Complex Conjugate
Polar Coordinates - Arg(z*)
Euler Formula

49. 2a
i²
z + z*
x-axis in the complex plane
natural

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

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CLEP General Mathematics: Complex Numbers

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