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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. Every complex number has the 'Standard Form':
|z-w|
a + bi for some real a and b.
Affix
Polar Coordinates - sin?

2. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
How to multiply complex nubers(2+i)(2i-3)
irrational
conjugate
the complex numbers

3. I^2 =
the complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Field
-1

4. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Euler's Formula
Roots of Unity
multiplying complex numbers
Polar Coordinates - sin?

5. Any number not rational
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Rules of Complex Arithmetic
irrational
Complex Subtraction

6. (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)
Complex Numbers: Add & subtract
How to find any Power
conjugate

7. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
i^2 = -1
Polar Coordinates - z?¹
(a + bi) = (c + bi) = (a + c) + ( b + d)i

8. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
irrational
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^3

9. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
ln z
Polar Coordinates - Multiplication
The Complex Numbers

10. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
transcendental
Rules of Complex Arithmetic
How to add and subtract complex numbers (2-3i)-(4+6i)

11. Written as fractions - terminating + repeating decimals
rational
adding complex numbers
i^2
Polar Coordinates - Division

12. Real and imaginary numbers
complex numbers
How to find any Power
multiplying complex numbers
interchangeable

13. Cos n? + i sin n? (for all n integers)
Polar Coordinates - sin?
Argand diagram
(cos? +isin?)n
zz*

14. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
v(-1)
Euler Formula
Imaginary number

15. The square root of -1.
z + z*
ln z
x-axis in the complex plane
Imaginary Unit

16. 1
i²
Integers
Complex Subtraction
the complex numbers

17. Derives z = a+bi
conjugate pairs
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^3
Euler Formula

18. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Euler's Formula
'i'
i^4

19. 1
i^0
Complex Number
i^3
i^2

20. z1z2* / |z2|²
Polar Coordinates - z
z1 / z2
z1 ^ (z2)
Polar Coordinates - z?¹

21. Starts at 1 - does not include 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
natural
Polar Coordinates - z
Liouville's Theorem -

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

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23. (e^(-y) - e^(y)) / 2i = i sinh y
Polar Coordinates - cos?
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Multiplication by i
sin iy

24. Multiply moduli and add arguments
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Multiplication
transcendental
|z| = mod(z)

25. 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.
cos iy
Real and Imaginary Parts
Complex numbers are points in the plane
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

26. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
Affix
Polar Coordinates - Division
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

27. (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
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
i^0
rational

28. 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
Any polynomial O(xn) - (n > 0)
Euler Formula
We say that c+di and c-di are complex conjugates.
v(-1)

29. The field of all rational and irrational numbers.
z1 / z2
irrational
the complex numbers
Real Numbers

30. When two complex numbers are multipiled together.
0 if and only if a = b = 0
Integers
Complex numbers are points in the plane
Complex Multiplication

31. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Polar Coordinates - r
radicals
Imaginary Unit

32. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Any polynomial O(xn) - (n > 0)
Argand diagram
The Complex Numbers
z1 ^ (z2)

33. When two complex numbers are added together.
Subfield
Imaginary Unit
Complex Numbers: Add & subtract
Complex Addition

34. 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
subtracting complex numbers
Complex Addition
|z| = mod(z)
i^0

35. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
The Complex Numbers
How to find any Power
Integers
Absolute Value of a Complex Number

36. A² + b² - real and non negative
Euler Formula
i^2 = -1
transcendental
zz*

37. Root negative - has letter i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
imaginary
has a solution.
adding complex numbers

38. Imaginary number

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39. A number that can be expressed as a fraction p/q where q is not equal to 0.
Affix
complex numbers
sin z
Rational Number

40. R?¹(cos? - isin?)
Polar Coordinates - z?¹
Polar Coordinates - z
subtracting complex numbers
natural

41. 2nd. Rule of Complex Arithmetic

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42. 2ib
z - z*
How to solve (2i+3)/(9-i)
irrational
Euler Formula

43. A complex number may be taken to the power of another complex number.
Complex Exponentiation
i^2 = -1
Polar Coordinates - sin?
Polar Coordinates - z

44. ½(e^(-y) +e^(y)) = cosh y
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z
i^3
cos iy

45. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
x-axis in the complex plane
i^3
i^1

46. No i
Integers
real
Real Numbers
Complex Number Formula

47. A complex number and its conjugate
Euler's Formula
conjugate pairs
the vector (a -b)
Subfield

48. A+bi
0 if and only if a = b = 0
Complex Number Formula
Complex Number
Complex Conjugate

49. 1st. Rule of Complex Arithmetic
Polar Coordinates - Multiplication by i
i^2 = -1
Imaginary Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

50. xpressions such as ``the complex number z'' - and ``the point z'' are now
Complex Multiplication
interchangeable
rational
multiply the numerator and the denominator by the complex conjugate of the denominator.