The Imaginary Unit, i Trivia Quiz

Answer: A solution of the equation x^2 = -1

Solving the equation x^2 = -1, one gets the values of "x" to be sqrt[-1] and -sqrt[-1] (i.e. the positive and negative square roots of -1 respectively). We denote either of the two solutions by the symbol "i", and thus get the two solutions of the equation to be i and -i.

So now that we've found that the equation x^2 = -1 has two solutions (i and -i), which is the real "i"? The definition itself of the imaginary unit ("a solution of the equation x^2 = -1") is ambiguous, because the equation has two solutions. However, all ambiguity can be removed if you just choose one of the two solutions and forever call it "the positive i".

Summing things up,
- The square root of -1 is an imaginary number denoted by the symbol "i".
- i squared equals -1.

Answer: It can't be plotted anywhere

The number i cannot be plotted on the real number line, because it isn't "real"! On the contrary, it is "imaginary", which means it can't be represented anywhere on the real number line.

However, there is an "imaginary number line", where all multiples of i- not necessary integral ones- can be represented. These numbers are called "imaginary numbers", and are, by definition, numbers whose square is less than or equal to zero. Examples of numbers that can be plotted on such a line are:
9.5223i,
-8.4i,
17i,
(pi)*i,
i,
-12.825790894289379377774i, etc.

All numbers represented on this line are imaginary; real numbers can't be plotted here. One notable exception to this rule is 0i: 0i is equal to just 0, which is considered BOTH a real and an imaginary number. (Don't forget that any number, when multiplied by 0, yields 0 as the product.)

The imaginary number line finds a place in the complex number plane, also known as an Argand Diagram (which you would have encountered in this quiz), where it plays the role of the y-axis.

Answer: 1

The eighth power of i,

i^8 = i^2 * i^2 * i^2 * i^2 = (-1 * -1) * (-1 * -1) = 1 * 1 = 1

Thus, i^8 = 1.

Once again, let's take a look at the integral powers of i:
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
i^5 = i
i^6 = -1
i^7 = -i
i^8 = 1
i^9 = i
i^10 = -1
i^11 = -i
i^12 = 1
You might have noticed that the powers of i form a cyclic pattern; the sequence (i, -1, -i, 1) keeps getting repeated.

Thus, we can frame the following rules; where "n" is an integer greater than or equal to zero:
i^(4n + 1) = i
i^(4n + 2) = - 1
i^(4n + 3) = - i
i^(4n) = 1

Simplifying things a bit: If you want to find the value of any integral power of i, first divide the exponent by 4.
- If the remainder is 0, the answer is 1 (i^0).
- If the remainder is 1, the answer is i (i^1).
- If the remainder is 2, the answer is -1 (i^2).
- If the remainder is 3, the answer is -i (i^3).

e.g. To find i^562, we first divide 562 by 4. The remainder is 2, which means the answer is i^2 = -1.

Answer: The formula sqrt[xy] = sqrt[x] * sqrt[y] does not hold when "x" and "y" are both negative real numbers.

The formula sqrt[xy] = sqrt[x] * sqrt[y], which you probably learnt in your early high-school years, is correct only when "x" and/or "y" is/are positive. Euler published a computation similar to this one in 1770, when the theory of complex numbers was still in its youth.

Answer: e^[-(pi)/2]

We have:

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

Raising both sides to the power i,

[e^{i*(pi)/2}]^i = i^i
=> i^i = e^[i*i*(pi)/2]
=> i^i = e^[-1*(pi)/2]
=> i^i = e^[-(pi)/2]

This, when solved, yields a remarkable result:

i^i = 0.2078795763...

So even though i by itself is not "real", and finds no place on the real number line, one can actually obtain the value of (i^i) ... amazing, isn't it?

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Euler's identity, which has been described as "the most remarkable formula in mathematics" by Richard Feynman, is the following equation and is true by definition:

e^[i*(pi)] + 1 = 0

Feynman describes this identity in such a manner because it contains the numbers e, pi, 1, 0 and the imaginary unit, as well as the basic operators of mathematics: exponentiation, multiplication, addition and equality.

Euler's identity can be easily derived by substituting x = pi in Euler's formula:

e^[i.x] = cos(x) + i.sin(x)
=> e^[i*(pi)] = cos(pi) + i.sin(pi)
=> e^[i*(pi)] = -1 + i.0
=> e^[i*(pi)] + 1 = 0

Answer: [plus or minus] (i + 1)/sqrt[2]

So far, what we have obtained is:

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

Notice that (1^2 + 2i + i^2) is of the form (a^2 + 2ab + b^2), where a = 1 and b = i. This can be simplified to give (i + 1)^2.

=> i = [(i + 1)^2]/2

Taking the square root of both sides,

sqrt[i] = [plus or minus] sqrt[(i+1)^2] / sqrt[2]
= [plus or minus] (i + 1)/sqrt[2]

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This result can also be easily obtained from Euler's formula, by substituting x = (pi)/2 in the formula and then taking the square root of both sides.

Answer: To avoid confusion with electric current, traditionally denoted by "I"

Electric current is defined as the rate of flow of charges, usually through an electrical conductor such as a copper wire.

The symbol used to denote electric current is the capital letter "I", from the German word "Intensität", which means "intensity". In mathematics, the symbol "i" was first introduced by Euler in 1748, possibly because "i" is the first letter of the Latin word "Imaginarius". (No prizes for guessing what that means!)

Answer: Yes

(-2.5i)^2
= -2.5i * -2.5i
= -2.5 * -2.5 * i * i
= 6.25 * i * i
= 6.25 * -1
= -6.25

Thus, as we see, the square of -2.5i is indeed negative, and therefore it IS imaginary. As a matter of fact, ANY multiple of i, whether its coefficient is positive or negative or zero, is an imaginary number.

The term "real number" was, it may be surprising to learn, coined only as a retronym in response to "imaginary number". Both real numbers and imaginary numbers are subsets of the set of complex numbers.

Answer: 6i and -6i

Let's first find the squares of 6i and -6i:

(6i)^2 = 6i * 6i = 36 * -1 = -36
(-6i)^2 = -6i * -6i = 36 * -1 = -36

Thus, as you can see, both 6i and -6i are square roots of -36. Similarly, 10i and -10i are the square roots of -100, and so on.

Answer: 0

The number 56.9 can be written as (56.9 + i0); thus, the imaginary part of the number is 0.

The complex numbers are merely and extension of the real numbers; besides real numbers, the set of complex numbers also includes imaginary numbers (like i, -86i, 14.4i, etc.).

As said before, all complex numbers can be represented in the form (a + ib), where both a and b are real numbers. For example,

(12 - i7): here, the real part = 12 and the imaginary part = -7

66i = (0 + i66): here, the real part = 0 and the imaginary part = 66

19 = (19 + i0): here, the real part = 19 and the imaginary part = 0

Answer: 2

Let's begin solving the given equation.

(x + i)(x - i) = 5
=> x^2 - i^2 = 5

Since i^2 = -1,

x^2 - (-1) = 5
=> x^2 + 1 = 5
=> x^2 = 5 - 1
=> x^2 = 4
=> x = [plus or minus] sqrt[4]
=> x = [plus or minus] 2

Since it is given that x is positive, its value is 2.

Answer: i

Substituting x = pi/4 and n = 2 in de Moivre's formula,

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

Now, cos(pi/2) = 0 and sin(pi/2) = 1. Therefore,

[cos(pi/4) + i.sin(pi/4)]^2
= 0 + i.1
= i

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De Moivre's formula can be derived from Euler's formula; however, it is worth noting that de Moivre's formula historically preceded Euler's formula. This formula can be easily proven for natural numbers by the principle of mathematical induction.

Answer: (-10,18)

The complex number z = [-2(5 - i9)] can be written as (-10 + i18), after opening the bracket. Thus, z is represented on the Argand Diagram by the point (-5,9).

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Let's now take a look at a real number, say, 8, which can be written as (8 + i0). It is represented on the Argand Diagram by the point (8,0), which, you will notice, is on the x-axis (the real axis). Similarly all other real numbers find their respective places on the x-axis of the Argand Diagram.

Now, let's take an example of an imaginary number, 21i. This can be written as (0 + i21), and hence is represented by the point (0,21). This point, like all the other points representing purely imaginary numbers, is located on the y-axis (the imaginary axis).

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The Argand Diagram was invented in 1806 by Jean-Robert Argand, a non-professional mathematician, while managing a book-store in Paris.

Answer: 5

The complex number z = (3 - i4) is represented by the point (3,-4) on the Argand Diagram.

Using Pythagoras' Theorem, the distance from the origin to the point (3,-4)
= sqrt[(3)^2 + (-4)^2]
= sqrt[9 + 16]
= sqrt[25]
= 5

Thus, |z| = 5.

Answer: Leonhard Euler

Despite being called "imaginary", imaginary numbers are just as "real" as the so-called "real numbers":

* Fractions such as 2/3 and 1/4 are meaningless to somebody counting, say, the number of number of people in a room; yet, they make sense to someone who is trying to cut a pie into pieces in a given ratio of sizes.

* Negative numbers like -453.6 and -21 don't make sense while talking about the age of a person, but have a large number of applications when dealing with money, in contexts such as debt, the change in the value of a share, etc.

Similar is the case with imaginary numbers- for most human tasks, they have no meaning, but they have concrete applications in various sciences and related fields, such as signal-processing, electromagnetism, quantum mechanics and cartography.

I'll illustrate this with an example: in electrical engineering, the values of electric current and voltage are sometimes expressed as imaginary numbers or complex numbers with non-zero imaginary parts, called "phasors". Even though these currents are supposedly "imaginary" (from the mathematical perspective), they can cause _real_ harm to people or damage to equipment!

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I hope you found this quiz entertaining as well as educational; feel free to send me a PM if you have any comments/feedback/suggestions.