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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. I
z1 ^ (z2)
We say that c+di and c-di are complex conjugates.
i^1
Rules of Complex Arithmetic

2. 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.
radicals
How to find any Power
Affix
Polar Coordinates - Multiplication

3. I^2 =
sin iy
z1 ^ (z2)
zz*
-1

4. Written as fractions - terminating + repeating decimals
integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
rational
Square Root

5. A complex number and its conjugate
Polar Coordinates - Multiplication by i
adding complex numbers
i^2
conjugate pairs

6. When two complex numbers are divided.
point of inflection
Complex Division
sin iy
complex numbers

7. 2nd. Rule of Complex Arithmetic

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8. The complex number z representing a+bi.
Imaginary Unit
the complex numbers
Affix
Complex Numbers: Add & subtract

9. 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 Number
Real and Imaginary Parts
i^2
Polar Coordinates - Multiplication

10. We can also think of the point z= a+ ib as
(a + bi) = (c + bi) = (a + c) + ( b + d)i
conjugate pairs
i^2 = -1
the vector (a -b)

11. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiply the numerator and the denominator by the complex conjugate of the denominator.
sin iy
multiplying complex numbers
four different numbers: i - -i - 1 - and -1.

12. A+bi
Complex Number Formula
complex
Polar Coordinates - z?¹
irrational

13. Equivalent to an Imaginary Unit.
Imaginary number
ln z
x-axis in the complex plane
Complex Numbers: Multiply

14. 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.
Complex Numbers: Multiply
subtracting complex numbers
Euler Formula
four different numbers: i - -i - 1 - and -1.

15. 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
Complex Subtraction
non-integers
We say that c+di and c-di are complex conjugates.
imaginary

16. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
Complex Multiplication
Complex Exponentiation
Polar Coordinates - Arg(z*)

17. All the powers of i can be written as
the complex numbers
Irrational Number
complex numbers
four different numbers: i - -i - 1 - and -1.

18. Every complex number has the 'Standard Form':
Argand diagram
a + bi for some real a and b.
|z-w|
Polar Coordinates - z?¹

19. 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
i^4
multiply the numerator and the denominator by the complex conjugate of the denominator.
interchangeable

20. 1st. Rule of Complex Arithmetic
Complex Numbers: Multiply
i^2 = -1
How to solve (2i+3)/(9-i)
sin z

21. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
(a + c) + ( b + d)i
Imaginary Unit
Complex numbers are points in the plane
Complex Numbers: Add & subtract

22. Given (4-2i) the complex conjugate would be (4+2i)
0 if and only if a = b = 0
conjugate pairs
Complex Conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

23. x + iy = r(cos? + isin?) = re^(i?)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z
(a + c) + ( b + d)i
i^2 = -1

24. To simplify a complex fraction
cosh²y - sinh²y
Imaginary number
z1 / z2
multiply the numerator and the denominator by the complex conjugate of the denominator.

25. V(x² + y²) = |z|
Polar Coordinates - r
transcendental
Rational Number
Irrational Number

26. To simplify the square root of a negative number
non-integers
Complex Exponentiation
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to add and subtract complex numbers (2-3i)-(4+6i)

27. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
multiplying complex numbers
(cos? +isin?)n
Imaginary number

28. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Imaginary Unit
Field
cos iy
a + bi for some real a and b.

29. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Complex Conjugate
multiply the numerator and the denominator by the complex conjugate of the denominator.
Argand diagram

30. ½(e^(-y) +e^(y)) = cosh y
Complex Conjugate
four different numbers: i - -i - 1 - and -1.
cos iy
Field

31. Root negative - has letter i
How to solve (2i+3)/(9-i)
x-axis in the complex plane
Affix
imaginary

32. Rotates anticlockwise by p/2
z1 / z2
real
Polar Coordinates - Multiplication by i
the vector (a -b)

33. When two complex numbers are subtracted from one another.
ln z
has a solution.
Complex Subtraction
Imaginary Unit

34. V(zz*) = v(a² + b²)
the vector (a -b)
|z| = mod(z)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
adding complex numbers

35. xpressions such as ``the complex number z'' - and ``the point z'' are now
the distance from z to the origin in the complex plane
integers
interchangeable
x-axis in the complex plane

36. Starts at 1 - does not include 0
Square Root
can't get out of the complex numbers by adding (or subtracting) or multiplying two
natural
i^2 = -1

37. Not on the numberline
non-integers
Complex Exponentiation
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z-w|

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

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39. (e^(iz) - e^(-iz)) / 2i
cos z
sin z
The Complex Numbers
natural

40. Multiply moduli and add arguments
x-axis in the complex plane
Polar Coordinates - Multiplication
i^0
We say that c+di and c-di are complex conjugates.

41. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Liouville's Theorem -
Complex Subtraction
z - z*

42. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
has a solution.
Complex Exponentiation
'i'

43. E ^ (z2 ln z1)
z1 ^ (z2)
Polar Coordinates - Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i
The Complex Numbers

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

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45. When two complex numbers are added together.
the distance from z to the origin in the complex plane
Complex Addition
x-axis in the complex plane
conjugate pairs

46. A number that can be expressed as a fraction p/q where q is not equal to 0.
Argand diagram
sin iy
Imaginary Unit
Rational Number

47. When two complex numbers are multipiled together.
Complex Multiplication
irrational
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - cos?

48. Any number not rational
adding complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
Field
irrational

49. For real a and b - a + bi =
a + bi for some real a and b.
0 if and only if a = b = 0
Polar Coordinates - z?¹
Every complex number has the 'Standard Form': a + bi for some real a and b.

50. In this amazing number field every algebraic equation in z with complex coefficients
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
irrational
has a solution.
Polar Coordinates - z?¹

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CLEP General Mathematics: Complex Numbers

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