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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. R?¹(cos? - isin?)
Polar Coordinates - z?¹
Polar Coordinates - r
Complex Number Formula
Liouville's Theorem -

2. 1
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^2
z1 / z2
Complex Exponentiation

3. Derives z = a+bi
How to multiply complex nubers(2+i)(2i-3)
(a + c) + ( b + d)i
How to solve (2i+3)/(9-i)
Euler Formula

4. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
sin iy
cos iy
Polar Coordinates - z

5. The product of an imaginary number and its conjugate is
Polar Coordinates - r
Complex Division
a real number: (a + bi)(a - bi) = a² + b²
i^2

6. Any number not rational
Subfield
Imaginary number
irrational
Complex numbers are points in the plane

7. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Polar Coordinates - Multiplication
0 if and only if a = b = 0
transcendental
Absolute Value of a Complex Number

8. All the powers of i can be written as
Roots of Unity
Euler's Formula
four different numbers: i - -i - 1 - and -1.
e^(ln z)

9. (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
transcendental
How to solve (2i+3)/(9-i)
|z-w|
Complex Division

10. 1st. Rule of Complex Arithmetic
i^2 = -1
Argand diagram
real
Polar Coordinates - r

11. The field of all rational and irrational numbers.
z + z*
Polar Coordinates - sin?
z1 ^ (z2)
Real Numbers

12. 2ib
Real and Imaginary Parts
Complex Addition
imaginary
z - z*

13. 5th. Rule of Complex Arithmetic
subtracting complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 ^ (z2)
How to find any Power

14. 3
How to find any Power
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - z
i^3

15. When two complex numbers are subtracted from one another.
point of inflection
Euler Formula
-1
Complex Subtraction

16. 3rd. Rule of Complex Arithmetic
How to solve (2i+3)/(9-i)
For real a and b - a + bi = 0 if and only if a = b = 0
z - z*
|z| = mod(z)

17. Like pi
Polar Coordinates - Multiplication by i
subtracting complex numbers
Euler's Formula
transcendental

18. Numbers on a numberline
Liouville's Theorem -
(cos? +isin?)n
i^1
integers

19. 1
Polar Coordinates - z?¹
i²
Rules of Complex Arithmetic
zz*

20. Rotates anticlockwise by p/2
Rational Number
The Complex Numbers
Polar Coordinates - Multiplication by i
transcendental

21. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
sin iy
-1
transcendental

22. 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
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - r
Complex Subtraction

23. 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
subtracting complex numbers
the complex numbers
has a solution.
Complex Division

24. xpressions such as ``the complex number z'' - and ``the point z'' are now
Complex Multiplication
interchangeable
Imaginary Unit
conjugate pairs

25. A² + b² - real and non negative
Complex numbers are points in the plane
conjugate pairs
zz*
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

26. A number that cannot be expressed as a fraction for any integer.
i^4
Irrational Number
cos z
non-integers

27. Where the curvature of the graph changes
point of inflection
Complex Multiplication
i^1
i^4

28. The complex number z representing a+bi.
Polar Coordinates - Arg(z*)
cos iy
irrational
Affix

29. V(zz*) = v(a² + b²)
z1 ^ (z2)
Field
has a solution.
|z| = mod(z)

30. Starts at 1 - does not include 0
Every complex number has the 'Standard Form': a + bi for some real a and b.
Subfield
Polar Coordinates - z
natural

31. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
ln z
four different numbers: i - -i - 1 - and -1.
How to multiply complex nubers(2+i)(2i-3)

32. 2nd. Rule of Complex Arithmetic

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33. I
Polar Coordinates - Arg(z*)
v(-1)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rules of Complex Arithmetic

34. ½(e^(iz) + e^(-iz))
v(-1)
Liouville's Theorem -
cos z
multiplying complex numbers

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

36. 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
conjugate pairs
conjugate
Real and Imaginary Parts
multiply the numerator and the denominator by the complex conjugate of the denominator.

37. We see in this way that the distance between two points z and w in the complex plane is
Complex Number Formula
Every complex number has the 'Standard Form': a + bi for some real a and b.
|z-w|
Euler Formula

38. 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
Polar Coordinates - z?¹
-1
'i'
subtracting complex numbers

39. ? = -tan?
Square Root
Polar Coordinates - Arg(z*)
i^0
Polar Coordinates - Multiplication by i

40. 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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41. A+bi
conjugate pairs
rational
Complex Number Formula
imaginary

42. 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
Rules of Complex Arithmetic
'i'
Square Root

43. 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.
Polar Coordinates - r
Complex Numbers: Multiply
Complex numbers are points in the plane
Polar Coordinates - Multiplication by i

44. When two complex numbers are added together.
irrational
Complex Addition
Euler's Formula
multiplying complex numbers

45. We can also think of the point z= a+ ib as
Complex Conjugate
the vector (a -b)
(a + c) + ( b + d)i
x-axis in the complex plane

46. The square root of -1.
Absolute Value of a Complex Number
Complex Multiplication
Imaginary Numbers
Imaginary Unit

47. (a + bi)(c + bi) =
De Moivre's Theorem
Complex Subtraction
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i

48. When two complex numbers are multipiled together.
conjugate pairs
complex
integers
Complex Multiplication

49. No i
For real a and b - a + bi = 0 if and only if a = b = 0
Euler Formula
interchangeable
real

50. A subset within a field.
Imaginary Numbers
Subfield
e^(ln z)
Every complex number has the 'Standard Form': a + bi for some real a and b.

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CLEP General Mathematics: Complex Numbers

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