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The Imaginary Unit, i

Created by achernar

Fun Trivia : Quizzes : Specific Math Topics
The Imaginary Unit i game quiz
"The imaginary unit, i, may seem out-of-this-world, but it really isn't; take this quiz and test your familiarity with i and the wonderful world of complex numbers. Enjoy!"

15 Points Per Correct Answer - No time limit  



1. How is the imaginary unit, i, defined?
    The first person singular pronoun in the nominative case
    The value of the function f(x) = x^2 - 1 when x = -1
    A solution of the equation x^2 = -1
    The derivative of the function f(x) = x^2 - 1 at x = 0


2. Back in your school days, you were taught how to plot numbers on a number-line, even the more offbeat ones like -59/3 and the square root of two. Where on the real number line should i be plotted?
    Somewhere between -1 and -(infinity)
    Somewhere between -1 and 1
    Exactly at 0
    It can't be plotted anywhere


3. Let's examine the first few whole-number powers of i:
i^0 = 1; The zeroth power of any number other than zero is one.
i^1 = i; Any number raised to the first power is the number itself.
i^2 = -1; By definition, i^2 = -1.
i^3 = -i; i^3 = i(i^2) = i(-1) = -i
i^4 = 1; i^4 = i^2 * i^2 = -1 * -1 = 1

You must have got the hang of things now...so tell me, what is the value of i^8, i.e. the eighth power of i?
    -1
    1
    i
    -i


4. State the basic flaw in the logic that has been employed in producing the following result:

1 = sqrt[1] = sqrt[-1 * -1] = sqrt[-1] * sqrt[-1] = i * i = -1

[Note that "sqrt" denotes the operation "to take the non-negative square root of".]
    There is a subtle difference between (i * i) and i^2.
    In the imaginary world, all logic and mathematical principles lose ground; as a result, this result is perfectly valid.
    There is no flaw; -1 IS equal to 1, because they have the same moduli.
    The formula sqrt[xy] = sqrt[x] * sqrt[y] does not hold when "x" and "y" are both negative real numbers.


5. Euler's formula, named after the great 18th-century Swiss mathematician and physicist, Leonhard Euler (pronounced as you would "oiler"), states that for any real number "x",

e^(ix) = cos(x) + i*sin(x)

[e is the base of the natural logarithm; i is the imaginary unit (the a square root of -1); and sine and cosine, respectively abbreviated sin and cos, are trigonometric functions.]

Let's now substitute the value x = (pi)/2 in the above formula.

e^[i*(pi)/2] = cos[(pi)/2] + i*sin[(pi)/2]

Now, cos[(pi)/2] and sin[(pi)/2] are respectively equal to 0 and 1. Therefore,

e^[i*(pi)/2] = 0 + i.1
=> e^[i*(pi)/2] = i

From the expression that has been derived above, what is the value of i^i?
    e^[-(pi)/2]
    e^pi
    e^i
    e^[(pi)^2 / 4]


6. We know that i is the square root of -1 ... but what is the square root of i itself? I'll get you started:

i
= 2i / 2
= (2i + 1 - 1)/2
= (2i + 1 + i^2)/2
= (1 + 2i + i^2)/2

So you now know that i = (1 + 2i + i^2)/2

From this expression, what value of the square root of i can one obtain?
    [plus or minus] 1
    [plus or minus] (i - 2) * sqrt[2]
    [plus or minus] 1 / sqrt[2]
    [plus or minus] (i + 1)/sqrt[2]


7. Why, in electrical engineering and allied fields, is the imaginary unit denoted by the letter "j" (as opposed to "i")?
    Because "i" is liable to be confused with the numeral 1
    To avoid confusion with the element iodine, whose symbol is "I" and is widely used to make semi-conductors
    Because letters "d" to "i" of the alphabet have been exclusively reserved for symbols of newly-discovered elements
    To avoid confusion with electric current, traditionally denoted by "I"


8. An imaginary number is defined as a number whose square is less than or equal to zero. Thus, numbers like 8i and 17i would be "imaginary", because their respective squares, -64 and -289, are negative.

The squares of the imaginary numbers that I used as examples can be calculated as follows:
(8i)^2 = 8i * 8i = (8 * 8) * (i * i) = 64 * -1 = -64
(17i)^2 = 17i * 17i = (17 * 17) * (i * i) = 289 * -1 = -289

The number 0 is also imaginary, because 0^2 = 0. (By the way, 0 is the only number which is both real and imaginary.)

Is the number -2.5i imaginary?
    Yes
    No


9. What result(s) does one get by taking the square root of -36?
    6i and -6i
    -6i
    6i
    -6^i


10. The set of complex numbers is an extension of the set of real numbers, each element of which can be represented in the form (a + ib), where "a" and "b" are real numbers. Examples of complex numbers are (-3 + i5), (7.98 + i) and (1 - i16.3).

The letter "a" is used to denote the so-called "real part" of the complex number and "b" represents the "imaginary part". Thus, in the complex number (4.9 - i3), 4.9 is the real part and -3 is the imaginary part.

What is the imaginary part of the complex number 56.9?
    0
    1
    56.9 isn't a complex number.
    56.9i


11. If (x + i)(x - i) = 5, where i is the imaginary unit and "x" is positive and real, what is the value of x?
    5
    0
    1
    2


12. De Moivre's formula, developed by the French mathematician Abraham de Moivre (1667 - 1754), states that for any real number "x" and integer "n",

[cos(x) + i.sin(x)]^n = cos(nx) + i.sin(nx)

Using this formula, calculate the value of the following expression:

[cos(pi/4) + i.sin(pi/4)]^2

[pi is the smallest positive "x" for which sin(x) = 0; sin and cos are trigonometric functions.]
    1
    cos(pi/12)
    2i
    i


13. Complex numbers can be represented as points on a plane with the help of an Argand Diagram, which consists of an x and a y-axis. The x-axis is called the real axis and the y-axis is the imaginary axis. Complex numbers are assigned points on this plane in such a manner that there is a one-to-one correspondence between the set of complex numbers and the set of points on the plane; i.e., each point represents a unique complex number and each complex number is represented by a unique point.

This is accomplished in the following manner:
- The x co-ordinate of a point represents the real part of a complex number.
- The y co-ordinate of the point represents the imaginary part of the complex number.

Thus, the complex number (x + iy) is represented on the Argand Diagram by the point (x,y).

For example,
(5 + i6): (5,6)
(6 - i5): (6,-5)
3i: (0, 3)
-12: (-12,0)
0: (0,0)
-i: (0,-1)

What are the co-ordinates of the point on the Argand Diagram which represents the complex number z = [-2(5 - i9)]?
    (5,-9)
    (-5,-9)
    (-10,18)
    (10,18)


14. The modulus of the complex number z = x + iy is defined as the distance from the origin of the Argand Diagram to the point P(x,y), and is denoted by the symbol |z|.

What is the modulus of the complex number z = (3 - i4)?
    Answer: (A number; the distance from the origin to a point (x,y) = sqrt[x^2 + y^2])


15. René Descartes (1637) first coined the term "imaginary number" in a derogatory sense in his "La Geometrie", because such numbers were thought not to exist. Who was the first to introduce the symbol "i" to denote the square root of -1?
    Jean-Robert Argand
    Leonhard Euler
    Abraham de Moivre
    Herschel Pinkes Remochel Krustofski, a.k.a. "Krusty the Clown"


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Compiled May 25 13