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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. 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
Real and Imaginary Parts
four different numbers: i - -i - 1 - and -1.
complex numbers
Square Root

2. 2a
z + z*
i^4
How to multiply complex nubers(2+i)(2i-3)
Every complex number has the 'Standard Form': a + bi for some real a and b.

3. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Number
Argand diagram

4. The field of all rational and irrational numbers.
zz*
Polar Coordinates - cos?
'i'
Real Numbers

5. We can also think of the point z= a+ ib as
v(-1)
the vector (a -b)
imaginary
Imaginary Unit

6. No i
|z-w|
rational
Absolute Value of a Complex Number
real

7. 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
i^1
Polar Coordinates - Arg(z*)
Real Numbers
We say that c+di and c-di are complex conjugates.

8. Numbers on a numberline
integers
irrational
Polar Coordinates - cos?
complex

9. When two complex numbers are added together.
Complex Addition
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Real and Imaginary Parts
Euler's Formula

10. y / r
point of inflection
Polar Coordinates - sin?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Division

11. A² + b² - real and non negative
Polar Coordinates - Multiplication by i
zz*
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - z

12. Starts at 1 - does not include 0
Polar Coordinates - sin?
How to find any Power
natural
interchangeable

13. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - z?¹
The Complex Numbers
z - z*

14. For real a and b - a + bi =
can't get out of the complex numbers by adding (or subtracting) or multiplying two
0 if and only if a = b = 0
z1 / z2
Affix

15. 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
transcendental
z - z*
natural

16. To simplify the square root of a negative number
|z-w|
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
has a solution.
How to add and subtract complex numbers (2-3i)-(4+6i)

17. 2ib
transcendental
i²
Square Root
z - z*

18. 1
a real number: (a + bi)(a - bi) = a² + b²
i^4
Complex Conjugate
Any polynomial O(xn) - (n > 0)

19. A+bi
Euler's Formula
Complex Number Formula
How to solve (2i+3)/(9-i)
sin iy

20. We see in this way that the distance between two points z and w in the complex plane is
Imaginary number
conjugate pairs
Polar Coordinates - Division
|z-w|

21. 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
x-axis in the complex plane
Complex Numbers: Add & subtract
Complex Division
complex

22. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
the complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
has a solution.
Integers

23. All numbers
complex
transcendental
Complex Exponentiation
Real Numbers

24. The product of an imaginary number and its conjugate is
Polar Coordinates - Arg(z*)
the distance from z to the origin in the complex plane
a real number: (a + bi)(a - bi) = a² + b²
How to solve (2i+3)/(9-i)

25. A subset within a field.
Imaginary Numbers
Polar Coordinates - Arg(z*)
Subfield
Polar Coordinates - Multiplication

26. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
'i'
Integers
Field
sin z

27. 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
Rules of Complex Arithmetic
transcendental
standard form of complex numbers
zz*

28. R^2 = x
Imaginary Unit
Square Root
Affix
a real number: (a + bi)(a - bi) = a² + b²

29. Rotates anticlockwise by p/2
Complex Conjugate
Polar Coordinates - Multiplication by i
i²
i^0

30. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
real
v(-1)
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - sin?

31. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - cos?
We say that c+di and c-di are complex conjugates.
Euler's Formula

32. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

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33. When two complex numbers are multipiled together.
standard form of complex numbers
i^0
Polar Coordinates - z
Complex Multiplication

34. (a + bi) = (c + bi) =
i^3
z1 / z2
Polar Coordinates - Arg(z*)
(a + c) + ( b + d)i

35. 4th. Rule of Complex Arithmetic
i^2 = -1
Absolute Value of a Complex Number
conjugate pairs
(a + bi) = (c + bi) = (a + c) + ( b + d)i

36. 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
subtracting complex numbers
Integers
How to add and subtract complex numbers (2-3i)-(4+6i)
the complex numbers

37. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
(a + c) + ( b + d)i
radicals
Polar Coordinates - z

38. A number that can be expressed as a fraction p/q where q is not equal to 0.
has a solution.
Imaginary Numbers
Complex Division
Rational Number

39. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
|z-w|
v(-1)
ln z
multiplying complex numbers

40. 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
Complex Addition
point of inflection
adding complex numbers
Real Numbers

41. Every complex number has the 'Standard Form':
Complex Numbers: Multiply
Complex Conjugate
a + bi for some real a and b.
Argand diagram

42. (e^(iz) - e^(-iz)) / 2i
sin z
Complex Number
Roots of Unity
How to multiply complex nubers(2+i)(2i-3)

43. Derives z = a+bi
How to add and subtract complex numbers (2-3i)-(4+6i)
Euler Formula
i^1
complex numbers

44. ½(e^(iz) + e^(-iz))
i^3
How to solve (2i+3)/(9-i)
cos z
zz*

45. The complex number z representing a+bi.
four different numbers: i - -i - 1 - and -1.
Affix
Any polynomial O(xn) - (n > 0)
v(-1)

46. Equivalent to an Imaginary Unit.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex
v(-1)
Imaginary number

47. Where the curvature of the graph changes
-1
Complex Division
the vector (a -b)
point of inflection

48. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
Polar Coordinates - Multiplication
Complex Numbers: Multiply
Complex Exponentiation
sin z

49. 1st. Rule of Complex Arithmetic
i²
i^3
Argand diagram
i^2 = -1

50. R?¹(cos? - isin?)
Polar Coordinates - z?¹
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Integers
non-integers