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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. Have radical
the distance from z to the origin in the complex plane
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - sin?
radicals

2. Given (4-2i) the complex conjugate would be (4+2i)
Every complex number has the 'Standard Form': a + bi for some real a and b.
|z| = mod(z)
Complex Conjugate
How to add and subtract complex numbers (2-3i)-(4+6i)

3. I^2 =
Polar Coordinates - cos?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
-1
Roots of Unity

4. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - z?¹
i^2
multiplying complex numbers

5. Written as fractions - terminating + repeating decimals
rational
Complex Conjugate
i^0
Polar Coordinates - Division

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

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7. 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
Absolute Value of a Complex Number
imaginary
Rules of Complex Arithmetic
zz*

8. All the powers of i can be written as
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
four different numbers: i - -i - 1 - and -1.
Rules of Complex Arithmetic
(cos? +isin?)n

9. E ^ (z2 ln z1)
z - z*
Imaginary Numbers
Field
z1 ^ (z2)

10. A number that can be expressed as a fraction p/q where q is not equal to 0.
Integers
four different numbers: i - -i - 1 - and -1.
Complex Subtraction
Rational Number

11. Has exactly n roots by the fundamental theorem of algebra
Complex Conjugate
Euler Formula
Any polynomial O(xn) - (n > 0)
natural

12. Where the curvature of the graph changes
Polar Coordinates - cos?
subtracting complex numbers
a real number: (a + bi)(a - bi) = a² + b²
point of inflection

13. 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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14. ? = -tan?
Roots of Unity
Euler's Formula
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Arg(z*)

15. A² + b² - real and non negative
i^3
Euler Formula
zz*
z1 / z2

16. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
z - z*
Complex Number Formula
point of inflection
multiplying complex numbers

17. A subset within a field.
Affix
the vector (a -b)
conjugate pairs
Subfield

18. R?¹(cos? - isin?)
Polar Coordinates - z?¹
i^1
the vector (a -b)
Irrational Number

19. x + iy = r(cos? + isin?) = re^(i?)
subtracting complex numbers
Polar Coordinates - Arg(z*)
the distance from z to the origin in the complex plane
Polar Coordinates - z

20. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Real Numbers
Polar Coordinates - Division
conjugate
Polar Coordinates - Multiplication by i

21. 5th. Rule of Complex Arithmetic
Polar Coordinates - z?¹
cos z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i²

22. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Complex Addition
i²
|z-w|

23. Like pi
transcendental
Polar Coordinates - Division
subtracting complex numbers
'i'

24. Rotates anticlockwise by p/2
imaginary
irrational
z + z*
Polar Coordinates - Multiplication by i

25. 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
Polar Coordinates - Multiplication by i
zz*
irrational

26. When two complex numbers are subtracted from one another.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Subtraction
Complex Conjugate
adding complex numbers

27. To simplify a complex fraction
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Subtraction
multiply the numerator and the denominator by the complex conjugate of the denominator.

28. Equivalent to an Imaginary Unit.
Imaginary number
How to solve (2i+3)/(9-i)
Imaginary Numbers
four different numbers: i - -i - 1 - and -1.

29. Starts at 1 - does not include 0
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Addition
integers
natural

30. (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
z1 / z2
Polar Coordinates - Division
How to solve (2i+3)/(9-i)
Complex numbers are points in the plane

31. A complex number may be taken to the power of another complex number.
radicals
Complex Exponentiation
Polar Coordinates - sin?
Polar Coordinates - Division

32. z1z2* / |z2|²
Real and Imaginary Parts
sin iy
|z| = mod(z)
z1 / z2

33. y / r
Any polynomial O(xn) - (n > 0)
Argand diagram
Polar Coordinates - sin?
i^2

34. Root negative - has letter i
imaginary
zz*
Complex Numbers: Multiply
Rules of Complex Arithmetic

35. 3
Field
point of inflection
i^3
conjugate pairs

36. (a + bi) = (c + bi) =
Polar Coordinates - Multiplication
(a + c) + ( b + d)i
interchangeable
z - z*

37. Cos n? + i sin n? (for all n integers)
Imaginary number
(cos? +isin?)n
0 if and only if a = b = 0
Polar Coordinates - Division

38. 2a
subtracting complex numbers
irrational
z + z*
Every complex number has the 'Standard Form': a + bi for some real a and b.

39. For real a and b - a + bi =
complex numbers
adding complex numbers
We say that c+di and c-di are complex conjugates.
0 if and only if a = b = 0

40. 1
We say that c+di and c-di are complex conjugates.
Imaginary Numbers
cosh²y - sinh²y
the vector (a -b)

41. V(zz*) = v(a² + b²)
z1 ^ (z2)
|z| = mod(z)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Euler's Formula

42. (e^(iz) - e^(-iz)) / 2i
sin z
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication by i
De Moivre's Theorem

43. 1
For real a and b - a + bi = 0 if and only if a = b = 0
transcendental
i²
Affix

44. The reals are just the
x-axis in the complex plane
complex
Complex Division
Complex numbers are points in the plane

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

46. 1
the vector (a -b)
Imaginary Numbers
i^4
Absolute Value of a Complex Number

47. 2nd. Rule of Complex Arithmetic

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48. 1
Complex Subtraction
i^2
Polar Coordinates - sin?
rational

49. A + bi
Polar Coordinates - Arg(z*)
standard form of complex numbers
|z-w|
e^(ln z)

50. Derives z = a+bi
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Euler Formula
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - cos?