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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
radicals
x-axis in the complex plane
multiplying complex numbers
z1 ^ (z2)

2. 1
i^1
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^0
ln z

3. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
four different numbers: i - -i - 1 - and -1.
i^2
Imaginary number

4. (a + bi) = (c + bi) =
z + z*
Complex Number Formula
(a + c) + ( b + d)i
cos iy

5. y / r
Polar Coordinates - sin?
cos z
Rational Number
|z| = mod(z)

6. 2ib
'i'
Integers
z - z*
Absolute Value of a Complex Number

7. Written as fractions - terminating + repeating decimals
Complex Multiplication
|z-w|
Absolute Value of a Complex Number
rational

8. 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
Rules of Complex Arithmetic
Imaginary Numbers
We say that c+di and c-di are complex conjugates.

9. 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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10. I^2 =
v(-1)
-1
De Moivre's Theorem
Field

11. 1
v(-1)
Imaginary number
i^2
Complex numbers are points in the plane

12. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
conjugate
z1 ^ (z2)
complex numbers

13. Derives z = a+bi
conjugate pairs
Complex Multiplication
sin iy
Euler Formula

14. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
Complex Numbers: Add & subtract
i^2 = -1
the distance from z to the origin in the complex plane
Polar Coordinates - cos?

15. V(zz*) = v(a² + b²)
|z| = mod(z)
interchangeable
v(-1)
Complex Numbers: Multiply

16. Numbers on a numberline
integers
Rational Number
How to multiply complex nubers(2+i)(2i-3)
i^0

17. 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
Complex Number
Polar Coordinates - z?¹
-1
the complex numbers

18. For real a and b - a + bi =
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
0 if and only if a = b = 0
radicals
rational

19. Starts at 1 - does not include 0
Argand diagram
natural
The Complex Numbers
i^1

20. When two complex numbers are multipiled together.
Complex Multiplication
point of inflection
complex
Rational Number

21. To simplify the square root of a negative number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Multiplication by 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)

22. We can also think of the point z= a+ ib as
can't get out of the complex numbers by adding (or subtracting) or multiplying two
point of inflection
Polar Coordinates - sin?
the vector (a -b)

23. The product of an imaginary number and its conjugate is
'i'
a real number: (a + bi)(a - bi) = a² + b²
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^0

24. (e^(-y) - e^(y)) / 2i = i sinh y
Complex Division
'i'
sin iy
zz*

25. Where the curvature of the graph changes
point of inflection
conjugate
has a solution.
cosh²y - sinh²y

26. x / r
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Real Numbers
Polar Coordinates - cos?
a + bi for some real a and b.

27. Rotates anticlockwise by p/2
Polar Coordinates - z?¹
Argand diagram
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Multiplication by i

28. z1z2* / |z2|²
0 if and only if a = b = 0
point of inflection
cosh²y - sinh²y
z1 / z2

29. Multiply moduli and add arguments
Polar Coordinates - Multiplication
Complex Numbers: Add & subtract
How to multiply complex nubers(2+i)(2i-3)
transcendental

30. All numbers
interchangeable
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
transcendental
complex

31. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
integers
conjugate
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Numbers: Add & subtract

32. No i
'i'
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^2
real

33. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Polar Coordinates - Multiplication
Real and Imaginary Parts
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

34. A + bi
Complex Numbers: Add & subtract
the vector (a -b)
cos z
standard form of complex numbers

35. 1st. Rule of Complex Arithmetic
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Integers
i^2 = -1
Complex Number Formula

36. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Polar Coordinates - z?¹
natural
the complex numbers

37. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication by i
How to add and subtract complex numbers (2-3i)-(4+6i)
sin z

38. Divide moduli and subtract arguments
The Complex Numbers
interchangeable
Real and Imaginary Parts
Polar Coordinates - Division

39. 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
Complex numbers are points in the plane
cos iy
Real and Imaginary Parts
rational

40. When two complex numbers are subtracted from one another.
Complex Subtraction
(a + c) + ( b + d)i
transcendental
Real Numbers

41. A² + b² - real and non negative
four different numbers: i - -i - 1 - and -1.
z1 / z2
zz*
i^2 = -1

42. A complex number and its conjugate
How to find any Power
e^(ln z)
'i'
conjugate pairs

43. ? = -tan?
Polar Coordinates - Arg(z*)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
complex
Polar Coordinates - Division

44. 2nd. Rule of Complex Arithmetic

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45. 1
cosh²y - sinh²y
Imaginary Unit
z + z*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

46. Equivalent to an Imaginary Unit.
We say that c+di and c-di are complex conjugates.
Imaginary number
v(-1)
rational

47. E^(ln r) e^(i?) e^(2pin)
multiplying complex numbers
e^(ln z)
How to multiply complex nubers(2+i)(2i-3)
a + bi for some real a and b.

48. The reals are just the
Imaginary Numbers
x-axis in the complex plane
Rules of Complex Arithmetic
the complex numbers

49. Root negative - has letter i
imaginary
-1
Complex Conjugate
sin iy

50. The complex number z representing a+bi.
Affix
real
i^3
irrational