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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. The reals are just the
i^1
real
x-axis in the complex plane
interchangeable

2. I
four different numbers: i - -i - 1 - and -1.
Complex Exponentiation
v(-1)
How to find any Power

3. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Real Numbers
Complex Number Formula
interchangeable

4. Written as fractions - terminating + repeating decimals
natural
rational
a real number: (a + bi)(a - bi) = a² + b²
Rational Number

5. V(x² + y²) = |z|
radicals
conjugate
Polar Coordinates - r
Imaginary Unit

6. (a + bi) = (c + bi) =
Polar Coordinates - z
(a + c) + ( b + d)i
0 if and only if a = b = 0
|z-w|

7. 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.'
e^(ln z)
Complex Number
complex numbers
Square Root

8. Has exactly n roots by the fundamental theorem of algebra
Integers
How to solve (2i+3)/(9-i)
Any polynomial O(xn) - (n > 0)
a real number: (a + bi)(a - bi) = a² + b²

9. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Complex Exponentiation
How to multiply complex nubers(2+i)(2i-3)
0 if and only if a = b = 0
How to add and subtract complex numbers (2-3i)-(4+6i)

10. 1
real
interchangeable
i^2
x-axis in the complex plane

11. ½(e^(-y) +e^(y)) = cosh y
cos iy
The Complex Numbers
non-integers
Imaginary Numbers

12. (a + bi)(c + bi) =
subtracting complex numbers
Imaginary Unit
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
has a solution.

13. A² + b² - real and non negative
i^3
zz*
cosh²y - sinh²y
the complex numbers

14. Where the curvature of the graph changes
Real and Imaginary Parts
Rules of Complex Arithmetic
'i'
point of inflection

15. A complex number and its conjugate
conjugate pairs
Imaginary number
Liouville's Theorem -
the complex numbers

16. We see in this way that the distance between two points z and w in the complex plane is
The Complex Numbers
|z-w|
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
multiplying complex numbers

17. To simplify a complex fraction
Polar Coordinates - sin?
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - cos?
conjugate

18. 2ib
z - z*
Complex Conjugate
multiplying complex numbers
zz*

19. To simplify the square root of a negative number
standard form of complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Multiply
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

20. 5th. Rule of Complex Arithmetic
Subfield
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Roots of Unity
Rules of Complex Arithmetic

21. 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
the complex numbers
|z| = mod(z)
Roots of Unity
-1

22. Root negative - has letter i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Integers
Field
imaginary

23. A + bi
Complex Number
sin iy
standard form of complex numbers
Complex Number Formula

24. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Argand diagram
(cos? +isin?)n
(a + c) + ( b + d)i

25. E ^ (z2 ln z1)
z1 ^ (z2)
|z| = mod(z)
subtracting complex numbers
four different numbers: i - -i - 1 - and -1.

26. I^2 =
-1
Absolute Value of a Complex Number
imaginary
Polar Coordinates - cos?

27. 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
Rules of Complex Arithmetic
the vector (a -b)
Complex Multiplication
Polar Coordinates - z?¹

28. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - z
complex numbers
Roots of Unity
transcendental

29. The field of all rational and irrational numbers.
cos z
Real Numbers
transcendental
real

30. Like pi
Euler Formula
i^2
Every complex number has the 'Standard Form': a + bi for some real a and b.
transcendental

31. 1
Polar Coordinates - z?¹
irrational
i²
How to solve (2i+3)/(9-i)

32. When two complex numbers are divided.
Complex Number Formula
Complex Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - cos?

33. R?¹(cos? - isin?)
conjugate
complex numbers
|z| = mod(z)
Polar Coordinates - z?¹

34. When two complex numbers are multipiled together.
Absolute Value of a Complex Number
z + z*
the complex numbers
Complex Multiplication

35. A plot of complex numbers as points.
i^3
Argand diagram
Affix
Imaginary number

36. Multiply moduli and add arguments
Complex Numbers: Add & subtract
Polar Coordinates - Multiplication
Polar Coordinates - Multiplication by i
Every complex number has the 'Standard Form': a + bi for some real a and b.

37. We can also think of the point z= a+ ib as
the vector (a -b)
i^3
Complex Numbers: Multiply
ln z

38. Any number not rational
Polar Coordinates - sin?
can't get out of the complex numbers by adding (or subtracting) or multiplying two
irrational
Complex numbers are points in the plane

39. All the powers of i can be written as
x-axis in the complex plane
cos z
four different numbers: i - -i - 1 - and -1.
cos iy

40. All numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
Argand diagram
complex
zz*

41. ? = -tan?
complex
the vector (a -b)
non-integers
Polar Coordinates - Arg(z*)

42. V(zz*) = v(a² + b²)
four different numbers: i - -i - 1 - and -1.
|z| = mod(z)
Every complex number has the 'Standard Form': a + bi for some real a and b.
adding complex numbers

43. (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
Polar Coordinates - z?¹
Complex Exponentiation
How to solve (2i+3)/(9-i)
Integers

44. 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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45. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Complex Number Formula
Complex Subtraction
standard form of complex numbers

46. 2nd. Rule of Complex Arithmetic

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47. When two complex numbers are subtracted from one another.
z1 ^ (z2)
imaginary
x-axis in the complex plane
Complex Subtraction

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

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49. (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
Imaginary Unit
How to multiply complex nubers(2+i)(2i-3)
Complex Addition
zz*

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

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CLEP General Mathematics: Complex Numbers

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