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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. I
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^3
multiply the numerator and the denominator by the complex conjugate of the denominator.
v(-1)

2. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Imaginary Unit
rational
How to solve (2i+3)/(9-i)

3. Equivalent to an Imaginary Unit.
transcendental
Euler Formula
adding complex numbers
Imaginary number

4. When two complex numbers are divided.
How to multiply complex nubers(2+i)(2i-3)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Division
point of inflection

5. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex
subtracting complex numbers

6. A number that can be expressed as a fraction p/q where q is not equal to 0.
Polar Coordinates - Multiplication by i
multiplying complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rational Number

7. Every complex number has the 'Standard Form':
transcendental
a + bi for some real a and b.
imaginary
Integers

8. 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
radicals
zz*
Complex Multiplication

9. Written as fractions - terminating + repeating decimals
Square Root
rational
point of inflection
Polar Coordinates - r

10. Not on the numberline
complex numbers
Polar Coordinates - Division
How to find any Power
non-integers

11. To simplify a complex fraction
Complex Conjugate
ln z
multiply the numerator and the denominator by the complex conjugate of the denominator.
a + bi for some real a and b.

12. To simplify the square root of a negative number
How to solve (2i+3)/(9-i)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary Numbers
De Moivre's Theorem

13. 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
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic
sin z
Polar Coordinates - r

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
multiply the numerator and the denominator by the complex conjugate of the denominator.
imaginary
z1 / z2

15. All the powers of i can be written as
Complex Multiplication
four different numbers: i - -i - 1 - and -1.
Rules of Complex Arithmetic
point of inflection

16. y / r
Polar Coordinates - sin?
Argand diagram
conjugate pairs
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

17. 1st. Rule of Complex Arithmetic
Imaginary number
Complex Numbers: Add & subtract
Polar Coordinates - Multiplication by i
i^2 = -1

18. ? = -tan?
Polar Coordinates - Arg(z*)
Complex Conjugate
How to find any Power
Polar Coordinates - Division

19. A number that cannot be expressed as a fraction for any integer.
ln z
Rational Number
Complex Conjugate
Irrational Number

20. A + bi
rational
Imaginary Numbers
Rational Number
standard form of complex numbers

21. 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.
i^1
Rules of Complex Arithmetic
e^(ln z)
How to find any Power

22. (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
Real Numbers
standard form of complex numbers
Subfield
How to solve (2i+3)/(9-i)

23. Divide moduli and subtract arguments
(a + c) + ( b + d)i
Polar Coordinates - Division
0 if and only if a = b = 0
Absolute Value of a Complex Number

24. 3
i^0
Irrational Number
i^3
Complex Division

25. Cos n? + i sin n? (for all n integers)
conjugate
(cos? +isin?)n
cos z
Imaginary Unit

26. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Complex Numbers: Multiply
integers
'i'

27. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Rational Number
0 if and only if a = b = 0
conjugate
cosh²y - sinh²y

28. Rotates anticlockwise by p/2
irrational
z - z*
Polar Coordinates - Multiplication by i
real

29. Given (4-2i) the complex conjugate would be (4+2i)
natural
Complex Conjugate
Complex Multiplication
z - z*

30. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
cosh²y - sinh²y
How to add and subtract complex numbers (2-3i)-(4+6i)
We say that c+di and c-di are complex conjugates.
ln z

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

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32. For real a and b - a + bi =
subtracting complex numbers
0 if and only if a = b = 0
Polar Coordinates - cos?
integers

33. I
the distance from z to the origin in the complex plane
i^1
(a + c) + ( b + d)i
z - z*

34. When two complex numbers are multipiled together.
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Multiplication by i
Complex Multiplication
Polar Coordinates - sin?

35. 1
Complex Exponentiation
Complex Number
Real Numbers
i²

36. E^(ln r) e^(i?) e^(2pin)
Rational Number
e^(ln z)
Complex Exponentiation
cos z

37. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
0 if and only if a = b = 0
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to add and subtract complex numbers (2-3i)-(4+6i)

38. The reals are just the
multiply the numerator and the denominator by the complex conjugate of the denominator.
Irrational Number
x-axis in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i

39. 2ib
a + bi for some real a and b.
z - z*
irrational
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

40. z1z2* / |z2|²
real
Integers
rational
z1 / z2

41. ½(e^(-y) +e^(y)) = cosh y
cos iy
standard form of complex numbers
0 if and only if a = b = 0
a real number: (a + bi)(a - bi) = a² + b²

42. R^2 = x
Complex Multiplication
How to solve (2i+3)/(9-i)
Square Root
Polar Coordinates - z

43. (e^(iz) - e^(-iz)) / 2i
'i'
Complex Division
has a solution.
sin z

44. I^2 =
cos iy
sin z
-1
a + bi for some real a and b.

45. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
transcendental
i^3
Complex Multiplication

46. A plot of complex numbers as points.
z1 ^ (z2)
Complex Multiplication
Argand diagram
natural

47. (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
Any polynomial O(xn) - (n > 0)
Field
z + z*
How to multiply complex nubers(2+i)(2i-3)

48. A² + b² - real and non negative
Complex numbers are points in the plane
zz*
the distance from z to the origin in the complex plane
a real number: (a + bi)(a - bi) = a² + b²

49. When two complex numbers are subtracted from one another.
a real number: (a + bi)(a - bi) = a² + b²
Real and Imaginary Parts
Imaginary Unit
Complex Subtraction

50. (a + bi)(c + bi) =
adding complex numbers
Complex Multiplication
|z| = mod(z)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i