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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. (e^(-y) - e^(y)) / 2i = i sinh y
Imaginary Numbers
We say that c+di and c-di are complex conjugates.
real
sin iy

2. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
real
i^1
e^(ln z)
Complex numbers are points in the plane

3. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
'i'
The Complex Numbers
zz*
Rational Number

4. 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
Polar Coordinates - z?¹
Affix
How to multiply complex nubers(2+i)(2i-3)
adding complex numbers

5. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
real
ln z
interchangeable
How to add and subtract complex numbers (2-3i)-(4+6i)

6. The product of an imaginary number and its conjugate is
transcendental
a real number: (a + bi)(a - bi) = a² + b²
i^2
radicals

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

8. 3
i^3
radicals
Polar Coordinates - sin?
standard form of complex numbers

9. E^(ln r) e^(i?) e^(2pin)
z + z*
e^(ln z)
i²
z1 ^ (z2)

10. 2nd. Rule of Complex Arithmetic

11. 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
the complex numbers
z + z*
subtracting complex numbers
Any polynomial O(xn) - (n > 0)

12. 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
Argand diagram
Real and Imaginary Parts
-1
a real number: (a + bi)(a - bi) = a² + b²

13. 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.'
a + bi for some real a and b.
Complex Number
(cos? +isin?)n
z + z*

14. 1
subtracting complex numbers
i^0
Roots of Unity
conjugate pairs

15. 2ib
radicals
z - z*
Roots of Unity
z1 / z2

16. We can also think of the point z= a+ ib as
z + z*
conjugate
the vector (a -b)
Polar Coordinates - sin?

17. 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
De Moivre's Theorem
subtracting complex numbers
|z| = mod(z)
We say that c+di and c-di are complex conjugates.

18. 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.
How to find any Power
Imaginary Unit
zz*
Liouville's Theorem -

19. ½(e^(-y) +e^(y)) = cosh y
non-integers
cos iy
standard form of complex numbers
complex numbers

20. The square root of -1.
Imaginary Unit
v(-1)
the vector (a -b)
i^4

21. A number that cannot be expressed as a fraction for any integer.
z1 / z2
Irrational Number
a real number: (a + bi)(a - bi) = a² + b²
Any polynomial O(xn) - (n > 0)

22. 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 distance from z to the origin in the complex plane
Real and Imaginary Parts
subtracting complex numbers
Polar Coordinates - r

23. A number that can be expressed as a fraction p/q where q is not equal to 0.
zz*
Rational Number
Complex Numbers: Multiply
i^0

24. 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.

25. 1
multiplying complex numbers
real
i^4
Polar Coordinates - Arg(z*)

26. V(zz*) = v(a² + b²)
Subfield
|z| = mod(z)
ln z
0 if and only if a = b = 0

27. When two complex numbers are subtracted from one another.
z1 / z2
Complex Subtraction
the complex numbers
complex numbers

28. When two complex numbers are added together.
Complex Addition
x-axis in the complex plane
cos iy
z1 / z2

29. 4th. Rule of Complex Arithmetic
v(-1)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos iy
Complex Subtraction

30. 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
-1
Complex Numbers: Add & subtract
the distance from z to the origin in the complex plane

31. Any number not rational
real
cos iy
irrational
0 if and only if a = b = 0

32. (e^(iz) - e^(-iz)) / 2i
sin z
Euler's Formula
(a + c) + ( b + d)i
Polar Coordinates - z

33. A + bi
Polar Coordinates - cos?
How to add and subtract complex numbers (2-3i)-(4+6i)
standard form of complex numbers
i^2

34. Multiply moduli and add arguments
Polar Coordinates - Multiplication
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Multiplication by i
Rational Number

35. A subset within a field.
Argand diagram
Real Numbers
Complex Exponentiation
Subfield

36. (a + bi) = (c + bi) =
non-integers
(a + c) + ( b + d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Square Root

37. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
|z| = mod(z)
the complex numbers
conjugate pairs

38. For real a and b - a + bi =
0 if and only if a = b = 0
sin iy
adding complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0

39. To simplify the square root of a negative number
Imaginary Unit
cos iy
How to multiply complex nubers(2+i)(2i-3)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

40. Has exactly n roots by the fundamental theorem of algebra
Rational Number
Any polynomial O(xn) - (n > 0)
How to add and subtract complex numbers (2-3i)-(4+6i)
Field

41. I^2 =
-1
cosh²y - sinh²y
Complex Numbers: Add & subtract
i^0

42. In this amazing number field every algebraic equation in z with complex coefficients
Complex Exponentiation
Imaginary Numbers
has a solution.
sin z

43. ½(e^(iz) + e^(-iz))
cos z
|z| = mod(z)
radicals
Polar Coordinates - z?¹

44. E ^ (z2 ln z1)
Complex Numbers: Multiply
z1 ^ (z2)
The Complex Numbers
Polar Coordinates - cos?

45. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
subtracting complex numbers
Absolute Value of a Complex Number
i²
z1 / z2

46. When two complex numbers are multipiled together.
zz*
Complex Multiplication
sin iy
-1

47. 1
Rules of Complex Arithmetic
conjugate pairs
i²
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

48. Equivalent to an Imaginary Unit.
De Moivre's Theorem
Irrational Number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Imaginary number

49. R^2 = x
natural
Square Root
Polar Coordinates - sin?
How to multiply complex nubers(2+i)(2i-3)

50. 3rd. Rule of Complex Arithmetic
subtracting complex numbers
i^2 = -1
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Conjugate