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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. Any number not rational
irrational
subtracting complex numbers
How to find any Power
Complex Exponentiation

2. 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.'
z + z*
imaginary
i^2
Complex Number

3. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Irrational Number
rational
i^2

4. I
a + bi for some real a and b.
i^1
complex
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

5. 1
i²
Polar Coordinates - Multiplication by i
0 if and only if a = b = 0
standard form of complex numbers

6. ? = -tan?
-1
transcendental
Real Numbers
Polar Coordinates - Arg(z*)

7. Real and imaginary numbers
standard form of complex numbers
complex numbers
Complex Division
imaginary

8. 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
Complex Multiplication
subtracting complex numbers
We say that c+di and c-di are complex conjugates.
cos iy

9. Rotates anticlockwise by p/2
Complex Exponentiation
Polar Coordinates - Multiplication by i
How to find any Power
Imaginary number

10. A complex number may be taken to the power of another complex number.
Polar Coordinates - sin?
Integers
Complex Exponentiation
interchangeable

11. All numbers
complex
Real and Imaginary Parts
radicals
Polar Coordinates - z

12. When two complex numbers are multipiled together.
0 if and only if a = b = 0
Complex Multiplication
i^1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

13. 3
Polar Coordinates - Arg(z*)
cos iy
i^3
Polar Coordinates - Multiplication by i

14. 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-w|
How to find any Power
Polar Coordinates - z?¹
Complex Addition

15. 1
Complex numbers are points in the plane
How to solve (2i+3)/(9-i)
Liouville's Theorem -
i^4

16. z1z2* / |z2|²
z - z*
Complex Addition
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z1 / z2

17. 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.

18. We see in this way that the distance between two points z and w in the complex plane is
|z| = mod(z)
a + bi for some real a and b.
four different numbers: i - -i - 1 - and -1.
|z-w|

19. Divide moduli and subtract arguments
Square Root
Polar Coordinates - Division
Euler Formula
non-integers

20. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Imaginary Numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
Real Numbers
Polar Coordinates - Arg(z*)

21. To simplify a complex fraction
Imaginary Unit
radicals
Every complex number has the 'Standard Form': a + bi for some real a and b.
multiply the numerator and the denominator by the complex conjugate of the denominator.

22. The product of an imaginary number and its conjugate is
has a solution.
a real number: (a + bi)(a - bi) = a² + b²
sin z
The Complex Numbers

23. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
a real number: (a + bi)(a - bi) = a² + b²
ln z
Polar Coordinates - Multiplication
How to add and subtract complex numbers (2-3i)-(4+6i)

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)
adding complex numbers
conjugate pairs
Polar Coordinates - z?¹

25. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
sin z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
standard form of complex numbers
Field

26. 5th. Rule of Complex Arithmetic
Affix
i^4
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 ^ (z2)

27. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Complex Number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
How to multiply complex nubers(2+i)(2i-3)

28. A number that cannot be expressed as a fraction for any integer.
natural
Complex Subtraction
Irrational Number
Complex Conjugate

29. 4th. Rule of Complex Arithmetic
Subfield
standard form of complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi) = (c + bi) = (a + c) + ( b + d)i

30. Has exactly n roots by the fundamental theorem of algebra
How to solve (2i+3)/(9-i)
conjugate
Complex Division
Any polynomial O(xn) - (n > 0)

31. The modulus of the complex number z= a + ib now can be interpreted as
De Moivre's Theorem
the distance from z to the origin in the complex plane
zz*
How to multiply complex nubers(2+i)(2i-3)

32. When two complex numbers are subtracted from one another.
z1 ^ (z2)
i²
real
Complex Subtraction

33. I
Absolute Value of a Complex Number
v(-1)
a + bi for some real a and b.
sin z

34. (e^(-y) - e^(y)) / 2i = i sinh y
e^(ln z)
z1 ^ (z2)
We say that c+di and c-di are complex conjugates.
sin iy

35. Every complex number has the 'Standard Form':
standard form of complex numbers
a + bi for some real a and b.
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Numbers: Add & subtract

36. Have radical
i^2
radicals
Square Root
Subfield

37. R?¹(cos? - isin?)
Complex Addition
Subfield
Polar Coordinates - z?¹
0 if and only if a = b = 0

38. 3rd. Rule of Complex Arithmetic
We say that c+di and c-di are complex conjugates.
non-integers
For real a and b - a + bi = 0 if and only if a = b = 0
Euler Formula

39. 1st. Rule of Complex Arithmetic
Roots of Unity
i^2 = -1
Field
Integers

40. R^2 = x
z - z*
Complex Division
Square Root
adding complex numbers

41. x + iy = r(cos? + isin?) = re^(i?)
How to multiply complex nubers(2+i)(2i-3)
rational
Liouville's Theorem -
Polar Coordinates - z

42. The field of all rational and irrational numbers.
Real Numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to solve (2i+3)/(9-i)
z1 ^ (z2)

43. 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: Add & subtract
standard form of complex numbers
Subfield
Imaginary Unit

44. I = imaginary unit - i² = -1 or i = v-1
i^2 = -1
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary Numbers
z1 / z2

45. For real a and b - a + bi =
0 if and only if a = b = 0
Complex Division
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)

46. 2ib
e^(ln z)
Rules of Complex Arithmetic
z - z*
z1 ^ (z2)

47. I^2 =
Polar Coordinates - z?¹
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
De Moivre's Theorem
-1

48. Derives z = a+bi
Subfield
Euler Formula
i^0
i^4

49. Multiply moduli and add arguments
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Real Numbers
Polar Coordinates - Multiplication
imaginary

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