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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. Numbers on a numberline
rational
integers
cosh²y - sinh²y
cos z

2. 2a
i^2
i²
non-integers
z + z*

3. Derives z = a+bi
Euler Formula
'i'
Polar Coordinates - r
Real Numbers

4. 1
natural
cosh²y - sinh²y
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i

5. A plot of complex numbers as points.
Polar Coordinates - Division
subtracting complex numbers
conjugate
Argand diagram

6. 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.
Complex Numbers: Multiply
Euler Formula
Polar Coordinates - z?¹
complex numbers

7. Root negative - has letter i
Imaginary number
imaginary
Complex Addition
the distance from z to the origin in the complex plane

8. 5th. Rule of Complex Arithmetic
transcendental
i^2
Field
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

9. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Square Root
Roots of Unity
e^(ln z)
v(-1)

10. The square root of -1.
Imaginary Unit
Roots of Unity
i^2 = -1
Absolute Value of a Complex Number

11. 1
i^2
Imaginary Unit
Absolute Value of a Complex Number
cos z

12. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Polar Coordinates - Arg(z*)
cos iy
ln z
0 if and only if a = b = 0

13. The product of an imaginary number and its conjugate is
Complex Addition
Complex Multiplication
Imaginary Unit
a real number: (a + bi)(a - bi) = a² + b²

14. When two complex numbers are multipiled together.
non-integers
v(-1)
x-axis in the complex plane
Complex Multiplication

15. 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
cos iy
point of inflection
For real a and b - a + bi = 0 if and only if a = b = 0
the complex numbers

16. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Real Numbers
Absolute Value of a Complex Number
z - z*
-1

17. 2nd. Rule of Complex Arithmetic

18. The reals are just the
x-axis in the complex plane
Complex Numbers: Add & subtract
rational
(a + bi) = (c + bi) = (a + c) + ( b + d)i

19. All numbers
-1
the vector (a -b)
complex
imaginary

20. 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
has a solution.
|z| = mod(z)
We say that c+di and c-di are complex conjugates.
e^(ln z)

21. 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.
Roots of Unity
How to find any Power
rational
non-integers

22. V(zz*) = v(a² + b²)
Complex Multiplication
|z| = mod(z)
Affix
subtracting complex numbers

23. Equivalent to an Imaginary Unit.
Polar Coordinates - z
Complex Conjugate
Complex Division
Imaginary number

24. Rotates anticlockwise by p/2
cos z
Polar Coordinates - Multiplication by i
standard form of complex numbers
z - z*

25. Divide moduli and subtract arguments
Polar Coordinates - cos?
Euler's Formula
Polar Coordinates - Division
Subfield

26. The field of all rational and irrational numbers.
Polar Coordinates - Multiplication
i^4
Real Numbers
the complex numbers

27. Multiply moduli and add arguments
i^2
Rules of Complex Arithmetic
Polar Coordinates - Multiplication
v(-1)

28. 1
How to add and subtract complex numbers (2-3i)-(4+6i)
i²
(cos? +isin?)n
z + z*

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

30. E ^ (z2 ln z1)
cos z
Rules of Complex Arithmetic
0 if and only if a = b = 0
z1 ^ (z2)

31. The modulus of the complex number z= a + ib now can be interpreted as
i^4
the distance from z to the origin in the complex plane
Polar Coordinates - z?¹
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

32. Any number not rational
irrational
Polar Coordinates - Multiplication
Complex Number Formula
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

33. Imaginary number

34. Real and imaginary numbers
Complex Division
complex numbers
Complex Exponentiation
Polar Coordinates - cos?

35. When two complex numbers are divided.
Subfield
four different numbers: i - -i - 1 - and -1.
Imaginary Unit
Complex Division

36. 3rd. Rule of Complex Arithmetic
real
Integers
standard form of complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0

37. In this amazing number field every algebraic equation in z with complex coefficients
We say that c+di and c-di are complex conjugates.
Complex Numbers: Multiply
sin z
has a solution.

38. A + bi
standard form of complex numbers
0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the complex numbers

39. z1z2* / |z2|²
z1 / z2
-1
Rules of Complex Arithmetic
i^3

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

41. I
has a solution.
i^1
rational
How to add and subtract complex numbers (2-3i)-(4+6i)

42. Not on the numberline
Complex numbers are points in the plane
How to add and subtract complex numbers (2-3i)-(4+6i)
non-integers
How to multiply complex nubers(2+i)(2i-3)

43. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
the vector (a -b)
Complex Subtraction
Integers
cos iy

44. 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.

45. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Real Numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
four different numbers: i - -i - 1 - and -1.
Rules of Complex Arithmetic

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

47. To simplify a complex fraction
Rules of Complex Arithmetic
The Complex Numbers
Real Numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.

48. 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
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
e^(ln z)
Rational Number

49. I^2 =
the distance from z to the origin in the complex plane
How to multiply complex nubers(2+i)(2i-3)
-1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

50. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Square Root
conjugate
Polar Coordinates - z?¹