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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.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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. For real a and b - a + bi =
Imaginary Unit
i²
0 if and only if a = b = 0
zz*

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

3. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - z?¹
four different numbers: i - -i - 1 - and -1.
Absolute Value of a Complex Number
e^(ln z)

4. 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.
|z| = mod(z)
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to find any Power
'i'

5. 2ib
transcendental
integers
z - z*
i^1

6. A + bi
cos iy
Subfield
standard form of complex numbers
Complex Numbers: Add & subtract

7. Divide moduli and subtract arguments
For real a and b - a + bi = 0 if and only if a = b = 0
cosh²y - sinh²y
Polar Coordinates - Division
Polar Coordinates - Multiplication

8. x / r
Polar Coordinates - cos?
How to find any Power
Subfield
non-integers

9. Not on the numberline
zz*
non-integers
Complex numbers are points in the plane
z1 ^ (z2)

10. (e^(iz) - e^(-iz)) / 2i
Polar Coordinates - Multiplication
v(-1)
sin z
Rules of Complex Arithmetic

11. Root negative - has letter i
imaginary
Absolute Value of a Complex Number
Complex numbers are points in the plane
integers

12. We can also think of the point z= a+ ib as
the vector (a -b)
Real Numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
rational

13. Numbers on a numberline
Euler's Formula
four different numbers: i - -i - 1 - and -1.
integers
rational

14. A+bi
Complex Number Formula
Polar Coordinates - Multiplication by i
a real number: (a + bi)(a - bi) = a² + b²
cosh²y - sinh²y

15. Multiply moduli and add arguments
Polar Coordinates - z?¹
Complex Exponentiation
Polar Coordinates - Multiplication
Every complex number has the 'Standard Form': a + bi for some real a and b.

16. The complex number z representing a+bi.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Division
Affix
conjugate pairs

17. Have radical
How to find any Power
sin z
radicals
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

18. 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.

19. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
transcendental
irrational
How to add and subtract complex numbers (2-3i)-(4+6i)
z - z*

20. When two complex numbers are divided.
i^3
real
non-integers
Complex Division

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

22. A complex number and its conjugate
Affix
conjugate pairs
Complex Numbers: Multiply
'i'

23. To simplify the square root of a negative number
Liouville's Theorem -
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
radicals
Polar Coordinates - sin?

24. All numbers
Field
complex
How to solve (2i+3)/(9-i)
|z| = mod(z)

25. 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
Rational Number
Complex Numbers: Multiply
interchangeable

26. Starts at 1 - does not include 0
Complex Division
a + bi for some real a and b.
natural
x-axis in the complex plane

27. 1
cosh²y - sinh²y
radicals
multiplying complex numbers
Complex Exponentiation

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

29. V(zz*) = v(a² + b²)
Complex Multiplication
Square Root
Euler's Formula
|z| = mod(z)

30. ½(e^(iz) + e^(-iz))
sin z
cos z
cosh²y - sinh²y
Polar Coordinates - Division

31. The modulus of the complex number z= a + ib now can be interpreted as
We say that c+di and c-di are complex conjugates.
the distance from z to the origin in the complex plane
Complex Subtraction
point of inflection

32. 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
Integers
We say that c+di and c-di are complex conjugates.
Any polynomial O(xn) - (n > 0)
real

33. 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
Complex Subtraction
Roots of Unity
interchangeable

34. I = imaginary unit - i² = -1 or i = v-1
i^3
a real number: (a + bi)(a - bi) = a² + b²
i^2
Imaginary Numbers

35. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
the vector (a -b)
z1 / z2
Complex Numbers: Add & subtract
subtracting complex numbers

36. R?¹(cos? - isin?)
Polar Coordinates - Multiplication
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - z?¹
(cos? +isin?)n

37. 1
i^4
Real and Imaginary Parts
imaginary
cos iy

38. Derives z = a+bi
Polar Coordinates - Arg(z*)
Polar Coordinates - r
complex numbers
Euler Formula

39. z1z2* / |z2|²
natural
0 if and only if a = b = 0
z1 / z2
We say that c+di and c-di are complex conjugates.

40. 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
Polar Coordinates - Multiplication by i
point of inflection
Roots of Unity

41. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
sin z
Absolute Value of a Complex Number
the vector (a -b)
Integers

42. I
Square Root
Euler's Formula
complex numbers
v(-1)

43. Where the curvature of the graph changes
i^2
Square Root
point of inflection
How to multiply complex nubers(2+i)(2i-3)

44. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
irrational
complex numbers
Roots of Unity
0 if and only if a = b = 0

45. The reals are just the
standard form of complex numbers
rational
the distance from z to the origin in the complex plane
x-axis in the complex plane

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

47. A number that cannot be expressed as a fraction for any integer.
x-axis in the complex plane
sin iy
Irrational Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

48. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Absolute Value of a Complex Number
ln z
Liouville's Theorem -
Imaginary Numbers

49. 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
Complex Numbers: Add & subtract
multiplying complex numbers
Any polynomial O(xn) - (n > 0)

50. 1st. Rule of Complex Arithmetic
How to add and subtract complex numbers (2-3i)-(4+6i)
sin z
a + bi for some real a and b.
i^2 = -1