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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. (a + bi) = (c + bi) =
How to add and subtract complex numbers (2-3i)-(4+6i)
z1 ^ (z2)
(a + c) + ( b + d)i
Polar Coordinates - Division

2. Root negative - has letter i
Complex Exponentiation
Integers
We say that c+di and c-di are complex conjugates.
imaginary

3. 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.
has a solution.
|z-w|
How to find any Power
subtracting complex numbers

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

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5. x / r
Complex numbers are points in the plane
irrational
Polar Coordinates - cos?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

6. Where the curvature of the graph changes
transcendental
point of inflection
conjugate pairs
Rational Number

7. 2a
the vector (a -b)
Argand diagram
z + z*
Rules of Complex Arithmetic

8. 1
i^0
zz*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
ln z

9. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
How to solve (2i+3)/(9-i)
transcendental
i^0

10. 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.
point of inflection
Argand diagram
has a solution.

11. Written as fractions - terminating + repeating decimals
Rules of Complex Arithmetic
non-integers
rational
Polar Coordinates - Multiplication

12. A number that cannot be expressed as a fraction for any integer.
Irrational Number
complex numbers
We say that c+di and c-di are complex conjugates.
How to multiply complex nubers(2+i)(2i-3)

13. A subset within a field.
Complex Number Formula
Subfield
Imaginary Unit
Complex Addition

14. The complex number z representing a+bi.
has a solution.
Affix
real
Euler's Formula

15. Like pi
transcendental
multiply the numerator and the denominator by the complex conjugate of the denominator.
subtracting complex numbers
Square Root

16. 4th. Rule of Complex Arithmetic
complex numbers
i^1
Polar Coordinates - Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i

17. 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.'
adding complex numbers
a real number: (a + bi)(a - bi) = a² + b²
i^2 = -1
Complex Number

18. To simplify the square root of a negative number
Complex Multiplication
Field
Complex Exponentiation
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

19. When two complex numbers are divided.
|z-w|
Complex Numbers: Multiply
Complex Number Formula
Complex Division

20. Not on the numberline
four different numbers: i - -i - 1 - and -1.
i^1
non-integers
Any polynomial O(xn) - (n > 0)

21. Real and imaginary numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
complex numbers
Absolute Value of a Complex Number
Imaginary Numbers

22. Imaginary number

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23. z1z2* / |z2|²
z1 / z2
Roots of Unity
Polar Coordinates - r
|z-w|

24. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
z1 / z2
integers
irrational

25. I^2 =
i^3
Rules of Complex Arithmetic
-1
complex

26. 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.
i^3
Subfield
How to multiply complex nubers(2+i)(2i-3)
Complex Numbers: Multiply

27. The reals are just the
The Complex Numbers
Square Root
standard form of complex numbers
x-axis in the complex plane

28. A complex number may be taken to the power of another complex number.
Complex Exponentiation
Absolute Value of a Complex Number
Square Root
Polar Coordinates - Division

29. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
(a + c) + ( b + d)i
has a solution.
x-axis in the complex plane
Absolute Value of a Complex Number

30. A complex number and its conjugate
multiplying complex numbers
Field
a real number: (a + bi)(a - bi) = a² + b²
conjugate pairs

31. ½(e^(iz) + e^(-iz))
z1 ^ (z2)
cos z
Complex Exponentiation
Polar Coordinates - z

32. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
(cos? +isin?)n
Any polynomial O(xn) - (n > 0)
the distance from z to the origin in the complex plane
Roots of Unity

33. R^2 = x
Real Numbers
has a solution.
Square Root
real

34. 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
De Moivre's Theorem
the complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
|z| = mod(z)

35. The modulus of the complex number z= a + ib now can be interpreted as
integers
Roots of Unity
Imaginary number
the distance from z to the origin in the complex plane

36. Numbers on a numberline
imaginary
cos iy
integers
z1 / z2

37. I = imaginary unit - i² = -1 or i = v-1
|z| = mod(z)
Imaginary Numbers
adding complex numbers
non-integers

38. 1
i²
Euler Formula
point of inflection
Real Numbers

39. E ^ (z2 ln z1)
z1 ^ (z2)
real
Complex Addition
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

40. 1st. Rule of Complex Arithmetic
Complex Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Absolute Value of a Complex Number
i^2 = -1

41. A² + b² - real and non negative
interchangeable
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Affix
zz*

42. 2nd. Rule of Complex Arithmetic

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43. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
adding complex numbers
|z-w|
i^2

44. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
non-integers
How to add and subtract complex numbers (2-3i)-(4+6i)
0 if and only if a = b = 0
For real a and b - a + bi = 0 if and only if a = b = 0

45. V(zz*) = v(a² + b²)
Roots of Unity
|z| = mod(z)
Polar Coordinates - Division
can't get out of the complex numbers by adding (or subtracting) or multiplying two

46. The product of an imaginary number and its conjugate is
Affix
four different numbers: i - -i - 1 - and -1.
Liouville's Theorem -
a real number: (a + bi)(a - bi) = a² + b²

47. Divide moduli and subtract arguments
Complex Number Formula
Euler's Formula
Polar Coordinates - Division
Imaginary Numbers

48. A+bi
Complex Number Formula
Polar Coordinates - cos?
complex numbers
Argand diagram

49. A number that can be expressed as a fraction p/q where q is not equal to 0.
How to add and subtract complex numbers (2-3i)-(4+6i)
Rational Number
ln z
Real and Imaginary Parts

50. R?¹(cos? - isin?)
Complex Subtraction
How to solve (2i+3)/(9-i)
'i'
Polar Coordinates - z?¹

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CLEP General Mathematics: Complex Numbers

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