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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. Like pi
Field
standard form of complex numbers
transcendental
Complex Division

2. When two complex numbers are divided.
Complex Division
cos iy
i^1
transcendental

3. I^2 =
Polar Coordinates - r
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Multiplication
-1

4. Equivalent to an Imaginary Unit.
i^0
i^1
i^2
Imaginary number

5. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
i^2
'i'
The Complex Numbers

6. All numbers
i²
i^1
complex
radicals

7. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Argand diagram
For real a and b - a + bi = 0 if and only if a = b = 0
i^0

8. 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
-1
Real and Imaginary Parts
Integers
Rules of Complex Arithmetic

9. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Real Numbers
Field
i^2 = -1
Polar Coordinates - Multiplication

10. Numbers on a numberline
Polar Coordinates - cos?
the vector (a -b)
integers
For real a and b - a + bi = 0 if and only if a = b = 0

11. 2ib
i^3
multiplying complex numbers
z - z*
Field

12. 2nd. Rule of Complex Arithmetic

13. I = imaginary unit - i² = -1 or i = v-1
conjugate pairs
i^0
Roots of Unity
Imaginary Numbers

14. 1
Rational Number
cosh²y - sinh²y
Complex Addition
|z-w|

15. Has exactly n roots by the fundamental theorem of algebra
i²
complex
cos iy
Any polynomial O(xn) - (n > 0)

16. A+bi
Complex Number Formula
cosh²y - sinh²y
Affix
Polar Coordinates - r

17. (a + bi) = (c + bi) =
standard form of complex numbers
Subfield
irrational
(a + c) + ( b + d)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. 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
Liouville's Theorem -
We say that c+di and c-di are complex conjugates.
a + bi for some real a and b.
Rational Number

20. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
adding complex numbers
sin z
Polar Coordinates - Division
The Complex Numbers

21. z1z2* / |z2|²
complex
z1 / z2
Complex Exponentiation
z - z*

22. 3rd. Rule of Complex Arithmetic
Complex Multiplication
adding complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
For real a and b - a + bi = 0 if and only if a = b = 0

23. We can also think of the point z= a+ ib as
i^2
x-axis in the complex plane
the vector (a -b)
For real a and b - a + bi = 0 if and only if a = b = 0

24. Starts at 1 - does not include 0
v(-1)
integers
natural
radicals

25. ½(e^(-y) +e^(y)) = cosh y
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos iy
Every complex number has the 'Standard Form': a + bi for some real a and b.
four different numbers: i - -i - 1 - and -1.

26. (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
a + bi for some real a and b.
the vector (a -b)
Polar Coordinates - z?¹
How to multiply complex nubers(2+i)(2i-3)

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

28. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
For real a and b - a + bi = 0 if and only if a = b = 0
How to add and subtract complex numbers (2-3i)-(4+6i)
Rational Number
Absolute Value of a Complex Number

29. No i
conjugate
real
Euler Formula
a + bi for some real a and b.

30. Where the curvature of the graph changes
Real and Imaginary Parts
i^0
point of inflection
Irrational Number

31. ? = -tan?
Polar Coordinates - Arg(z*)
v(-1)
complex
complex numbers

32. Divide moduli and subtract arguments
ln z
Complex numbers are points in the plane
Real Numbers
Polar Coordinates - Division

33. Real and imaginary numbers
complex numbers
Integers
transcendental
the vector (a -b)

34. (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 add and subtract complex numbers (2-3i)-(4+6i)
the complex numbers
How to solve (2i+3)/(9-i)
integers

35. 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
'i'
Affix
(a + bi) = (c + bi) = (a + c) + ( b + d)i

36. V(zz*) = v(a² + b²)
|z| = mod(z)
Roots of Unity
complex numbers
Field

37. ½(e^(iz) + e^(-iz))
sin iy
Euler Formula
cos z
conjugate

38. When two complex numbers are multipiled together.
standard form of complex numbers
i^4
x-axis in the complex plane
Complex Multiplication

39. 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
Polar Coordinates - sin?
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^2
subtracting complex numbers

40. For real a and b - a + bi =
i^2 = -1
multiplying complex numbers
0 if and only if a = b = 0
Argand diagram

41. 1
i^0
conjugate
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Division

42. 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.'
subtracting complex numbers
i^4
irrational
Complex Number

43. A subset within a field.
conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Any polynomial O(xn) - (n > 0)
Subfield

44. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Real and Imaginary Parts
sin iy
How to add and subtract complex numbers (2-3i)-(4+6i)
e^(ln z)

45. xpressions such as ``the complex number z'' - and ``the point z'' are now
i^4
interchangeable
four different numbers: i - -i - 1 - and -1.
How to multiply complex nubers(2+i)(2i-3)

46. Multiply moduli and add arguments
Polar Coordinates - Multiplication
How to multiply complex nubers(2+i)(2i-3)
sin iy
Polar Coordinates - Division

47. 2a
z + z*
non-integers
How to add and subtract complex numbers (2-3i)-(4+6i)
a real number: (a + bi)(a - bi) = a² + b²

48. E^(ln r) e^(i?) e^(2pin)
Imaginary Unit
Affix
e^(ln z)
point of inflection

49. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
conjugate
ln z
sin iy
a real number: (a + bi)(a - bi) = a² + b²

50. y / r
Polar Coordinates - sin?
Integers
Polar Coordinates - Multiplication
(a + c) + ( b + d)i