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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. 1
How to add and subtract complex numbers (2-3i)-(4+6i)
z - z*
cosh²y - sinh²y
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

2. A number that can be expressed as a fraction p/q where q is not equal to 0.
complex
Complex Numbers: Multiply
has a solution.
Rational Number

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

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4. A complex number and its conjugate
real
zz*
conjugate pairs
(a + bi) = (c + bi) = (a + c) + ( b + d)i

5. Multiply moduli and add arguments
rational
Liouville's Theorem -
'i'
Polar Coordinates - Multiplication

6. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
a real number: (a + bi)(a - bi) = a² + b²
|z| = mod(z)
Complex Subtraction

7. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate pairs
i^4
sin z
conjugate

8. Given (4-2i) the complex conjugate would be (4+2i)
the complex numbers
cos z
conjugate
Complex Conjugate

9. E ^ (z2 ln z1)
Argand diagram
z1 ^ (z2)
Absolute Value of a Complex Number
How to solve (2i+3)/(9-i)

10. For real a and b - a + bi =
0 if and only if a = b = 0
v(-1)
conjugate pairs
How to solve (2i+3)/(9-i)

11. To simplify the square root of a negative number
Complex Numbers: Add & subtract
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^2 = -1

12. 2nd. Rule of Complex Arithmetic

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13. A subset within a field.
Subfield
standard form of complex numbers
Complex Addition
the distance from z to the origin in the complex plane

14. (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
How to solve (2i+3)/(9-i)
complex
point of inflection
radicals

15. 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.'
Complex Division
Complex Number
Complex Number Formula
Complex numbers are points in the plane

16. Numbers on a numberline
integers
Polar Coordinates - sin?
i^0
Square Root

17. The modulus of the complex number z= a + ib now can be interpreted as
e^(ln z)
the distance from z to the origin in the complex plane
How to multiply complex nubers(2+i)(2i-3)
Field

18. 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
Complex Addition
z + z*
Euler's Formula

19. A number that cannot be expressed as a fraction for any integer.
zz*
i^0
-1
Irrational Number

20. (a + bi)(c + bi) =
integers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to add and subtract complex numbers (2-3i)-(4+6i)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

21. (e^(-y) - e^(y)) / 2i = i sinh y
i^2 = -1
sin iy
integers
radicals

22. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
Liouville's Theorem -
Real Numbers
Real and Imaginary Parts

23. 1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^0
Complex Numbers: Multiply
Complex Numbers: Add & subtract

24. ? = -tan?
Polar Coordinates - Arg(z*)
Imaginary number
Subfield
Complex Conjugate

25. A+bi
(cos? +isin?)n
Complex Number Formula
Complex Subtraction
z - z*

26. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
sin z
How to multiply complex nubers(2+i)(2i-3)
How to add and subtract complex numbers (2-3i)-(4+6i)
Roots of Unity

27. I^2 =
has a solution.
-1
i^4
sin z

28. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic
four different numbers: i - -i - 1 - and -1.

29. Have radical
cos z
For real a and b - a + bi = 0 if and only if a = b = 0
radicals
Complex numbers are points in the plane

30. 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
transcendental
i^2
Complex Number

31. A complex number may be taken to the power of another complex number.
ln z
Complex Exponentiation
Polar Coordinates - z
multiplying complex numbers

32. A plot of complex numbers as points.
How to find any Power
Argand diagram
We say that c+di and c-di are complex conjugates.
cos iy

33. When two complex numbers are subtracted from one another.
Complex Subtraction
point of inflection
Imaginary Numbers
cosh²y - sinh²y

34. R?¹(cos? - isin?)
Polar Coordinates - z?¹
point of inflection
Real and Imaginary Parts
i^2 = -1

35. The square root of -1.
Imaginary Unit
rational
We say that c+di and c-di are complex conjugates.
Euler's Formula

36. We can also think of the point z= a+ ib as
Argand diagram
the vector (a -b)
Polar Coordinates - Multiplication
Complex Subtraction

37. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
How to solve (2i+3)/(9-i)
radicals
The Complex Numbers
Complex numbers are points in the plane

38. The field of all rational and irrational numbers.
Real Numbers
i^4
(cos? +isin?)n
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

39. No i
sin z
real
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - r

40. 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
Real and Imaginary Parts
Imaginary Unit
(a + c) + ( b + d)i
Euler Formula

41. 1
i²
Liouville's Theorem -
z + z*
Polar Coordinates - r

42. All numbers
zz*
complex
conjugate pairs
Euler Formula

43. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
'i'
How to solve (2i+3)/(9-i)
Roots of Unity

44. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
real
i^2 = -1
integers

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

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46. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
rational
i²
Euler's Formula

47. Has exactly n roots by the fundamental theorem of algebra
(cos? +isin?)n
Any polynomial O(xn) - (n > 0)
(a + c) + ( b + d)i
ln z

48. Cos n? + i sin n? (for all n integers)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Numbers: Add & subtract
(cos? +isin?)n
Real Numbers

49. ½(e^(iz) + e^(-iz))
Euler's Formula
cosh²y - sinh²y
cos z
Complex Multiplication

50. 2a
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z + z*
Rational Number
How to find any Power

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CLEP General Mathematics: Complex Numbers

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