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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. z1z2* / |z2|²
z1 / z2
How to solve (2i+3)/(9-i)
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^2 = -1

2. (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
i^2
How to multiply complex nubers(2+i)(2i-3)
Absolute Value of a Complex Number
the complex numbers

3. 2nd. Rule of Complex Arithmetic

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4. A subset within a field.
Rules of Complex Arithmetic
-1
Subfield
Imaginary Numbers

5. 4th. Rule of Complex Arithmetic
Polar Coordinates - Division
cos z
zz*
(a + bi) = (c + bi) = (a + c) + ( b + d)i

6. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Rules of Complex Arithmetic
adding complex numbers
Absolute Value of a Complex Number

7. The complex number z representing a+bi.
Affix
multiplying complex numbers
Complex Multiplication
Euler's Formula

8. 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.
Absolute Value of a Complex Number
Complex numbers are points in the plane
How to find any Power
i^2

9. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
De Moivre's Theorem
Complex Numbers: Add & subtract
a real number: (a + bi)(a - bi) = a² + b²

10. Multiply moduli and add arguments
Irrational Number
Polar Coordinates - Multiplication
How to add and subtract complex numbers (2-3i)-(4+6i)
rational

11. Imaginary number

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12. Numbers on a numberline
can't get out of the complex numbers by adding (or subtracting) or multiplying two
integers
point of inflection
conjugate

13. The square root of -1.
Integers
i^4
i^1
Imaginary Unit

14. (e^(iz) - e^(-iz)) / 2i
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Conjugate
sin z
How to find any Power

15. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
i^2
imaginary
non-integers

16. Where the curvature of the graph changes
Complex Number
point of inflection
-1
z - z*

17. When two complex numbers are added together.
Polar Coordinates - Arg(z*)
i^0
z1 / z2
Complex Addition

18. 1
Complex Numbers: Multiply
cosh²y - sinh²y
Polar Coordinates - Division
-1

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

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20. I
irrational
z + z*
v(-1)
a real number: (a + bi)(a - bi) = a² + b²

21. 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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22. 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
'i'
Real Numbers
-1

23. (a + bi)(c + bi) =
How to multiply complex nubers(2+i)(2i-3)
Square Root
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number Formula

24. (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)
|z| = mod(z)
0 if and only if a = b = 0
Absolute Value of a Complex Number

25. 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
z - z*
standard form of complex numbers
Rules of Complex Arithmetic
Complex Numbers: Add & subtract

26. y / r
Subfield
Polar Coordinates - sin?
Imaginary number
ln z

27. ½(e^(iz) + e^(-iz))
Complex numbers are points in the plane
How to find any Power
i^4
cos z

28. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - sin?
Subfield
multiply the numerator and the denominator by the complex conjugate of the denominator.

29. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
i^2
0 if and only if a = b = 0
How to solve (2i+3)/(9-i)
multiplying complex numbers

30. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
subtracting complex numbers
cos z
Polar Coordinates - cos?

31. R?¹(cos? - isin?)
Imaginary number
rational
Polar Coordinates - z?¹
Rational Number

32. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
subtracting complex numbers
Polar Coordinates - sin?
ln z

33. Real and imaginary numbers
rational
complex numbers
Polar Coordinates - Multiplication by i
sin z

34. 1
i^2
Affix
Irrational Number
radicals

35. E ^ (z2 ln z1)
cos z
z + z*
For real a and b - a + bi = 0 if and only if a = b = 0
z1 ^ (z2)

36. Every complex number has the 'Standard Form':
a + bi for some real a and b.
Complex numbers are points in the plane
sin z
Euler Formula

37. Like pi
ln z
transcendental
Integers
How to add and subtract complex numbers (2-3i)-(4+6i)

38. For real a and b - a + bi =
0 if and only if a = b = 0
Euler's Formula
Subfield
Complex Conjugate

39. A² + b² - real and non negative
Real Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
zz*
0 if and only if a = b = 0

40. The product of an imaginary number and its conjugate is
irrational
The Complex Numbers
How to solve (2i+3)/(9-i)
a real number: (a + bi)(a - bi) = a² + b²

41. The reals are just the
point of inflection
Euler's Formula
x-axis in the complex plane
the complex numbers

42. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Complex Subtraction
How to add and subtract complex numbers (2-3i)-(4+6i)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
irrational

43. 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
Rules of Complex Arithmetic
Affix
De Moivre's Theorem
adding complex numbers

44. 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
Polar Coordinates - Division
We say that c+di and c-di are complex conjugates.
Polar Coordinates - z
Imaginary Unit

45. Equivalent to an Imaginary Unit.
Euler's Formula
Imaginary number
Subfield
Roots of Unity

46. All numbers
complex
Integers
0 if and only if a = b = 0
Polar Coordinates - cos?

47. Cos n? + i sin n? (for all n integers)
Complex numbers are points in the plane
Complex Addition
z + z*
(cos? +isin?)n

48. The modulus of the complex number z= a + ib now can be interpreted as
Subfield
Liouville's Theorem -
the distance from z to the origin in the complex plane
cosh²y - sinh²y

49. I^2 =
standard form of complex numbers
imaginary
-1
Complex Numbers: Multiply

50. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
rational
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary number