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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
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. Starts at 1 - does not include 0
Euler's Formula
Imaginary Numbers
natural
De Moivre's Theorem

2. No i
interchangeable
i^4
real
Complex Subtraction

3. 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.'
How to multiply complex nubers(2+i)(2i-3)
cos iy
standard form of complex numbers
Complex Number

4. ½(e^(iz) + e^(-iz))
cos z
Imaginary number
Polar Coordinates - Multiplication by i
i^3

5. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
subtracting complex numbers
We say that c+di and c-di are complex conjugates.
|z| = mod(z)

6. (a + bi) = (c + bi) =
cosh²y - sinh²y
Imaginary number
(a + c) + ( b + d)i
Complex Numbers: Add & subtract

7. 4th. Rule of Complex Arithmetic
rational
Complex Exponentiation
(a + bi) = (c + bi) = (a + c) + ( b + d)i
can't get out of the complex numbers by adding (or subtracting) or multiplying two

8. y / r
Polar Coordinates - sin?
Polar Coordinates - cos?
Affix
The Complex Numbers

9. When two complex numbers are subtracted from one another.
|z-w|
Complex Exponentiation
Complex Subtraction
natural

10. Root negative - has letter i
Complex Numbers: Add & subtract
imaginary
Integers
irrational

11. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Integers
subtracting complex numbers
i^2
How to add and subtract complex numbers (2-3i)-(4+6i)

12. 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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13. 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 - z?¹
zz*
We say that c+di and c-di are complex conjugates.
subtracting complex numbers

14. (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
sin iy
Real and Imaginary Parts
v(-1)
How to solve (2i+3)/(9-i)

15. 1st. Rule of Complex Arithmetic
has a solution.
point of inflection
complex numbers
i^2 = -1

16. Rotates anticlockwise by p/2
Imaginary Numbers
Polar Coordinates - Multiplication by i
i^2 = -1
Complex Multiplication

17. 1
standard form of complex numbers
cos z
i^2
i^1

18. Multiply moduli and add arguments
cosh²y - sinh²y
Polar Coordinates - Arg(z*)
Polar Coordinates - Multiplication
transcendental

19. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Integers
The Complex Numbers
Polar Coordinates - Division
e^(ln z)

20. x / r
Polar Coordinates - Multiplication
Polar Coordinates - cos?
z1 / z2
|z-w|

21. Derives z = a+bi
conjugate
Euler Formula
Real Numbers
The Complex Numbers

22. Numbers on a numberline
z1 ^ (z2)
radicals
integers
Polar Coordinates - sin?

23. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
i^0
Field
Euler Formula
Polar Coordinates - Multiplication

24. Divide moduli and subtract arguments
Imaginary Numbers
Polar Coordinates - Division
non-integers
has a solution.

25. 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
Liouville's Theorem -
interchangeable
Real and Imaginary Parts
Rational Number

26. I
Integers
i^2
Polar Coordinates - cos?
i^1

27. Equivalent to an Imaginary Unit.
complex
multiply the numerator and the denominator by the complex conjugate of the denominator.
Imaginary number
integers

28. xpressions such as ``the complex number z'' - and ``the point z'' are now
Imaginary Unit
Complex Subtraction
De Moivre's Theorem
interchangeable

29. I
v(-1)
Complex Exponentiation
Polar Coordinates - Multiplication by i
complex

30. 1
i^3
i^2 = -1
i^4
i²

31. A + bi
standard form of complex numbers
Rules of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Irrational Number

32. Have radical
The Complex Numbers
radicals
i²
Euler's Formula

33. 2ib
Imaginary number
Complex Exponentiation
Complex Numbers: Add & subtract
z - z*

34. When two complex numbers are divided.
Complex Division
imaginary
z1 ^ (z2)
Euler's Formula

35. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
i^4
ln z
cos iy
z1 ^ (z2)

36. 1
i²
Complex Addition
Square Root
multiplying complex numbers

37. Where the curvature of the graph changes
four different numbers: i - -i - 1 - and -1.
point of inflection
z + z*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

38. 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
real
adding complex numbers
a + bi for some real a and b.
the vector (a -b)

39. 1
Rules of Complex Arithmetic
Complex Numbers: Add & subtract
irrational
i^0

40. Not on the numberline
Polar Coordinates - Multiplication by i
non-integers
Complex Subtraction
irrational

41. In this amazing number field every algebraic equation in z with complex coefficients
Field
has a solution.
Any polynomial O(xn) - (n > 0)
Complex Numbers: Add & subtract

42. A number that can be expressed as a fraction p/q where q is not equal to 0.
Subfield
Rational Number
zz*
the complex numbers

43. Real and imaginary numbers
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - cos?
complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

44. A number that cannot be expressed as a fraction for any integer.
Irrational Number
conjugate
non-integers
radicals

45. Like pi
Rational Number
0 if and only if a = b = 0
transcendental
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

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

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47. 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
cos iy
Complex Numbers: Add & subtract
Complex Numbers: Multiply
How to solve (2i+3)/(9-i)

48. The product of an imaginary number and its conjugate is
a + bi for some real a and b.
subtracting complex numbers
a real number: (a + bi)(a - bi) = a² + b²
real

49. Has exactly n roots by the fundamental theorem of algebra
Real and Imaginary Parts
the vector (a -b)
Any polynomial O(xn) - (n > 0)
Real Numbers

50. A complex number may be taken to the power of another complex number.
Complex Exponentiation
(cos? +isin?)n
-1
irrational