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Quiz about The Imaginary Unit i
Quiz about The Imaginary Unit i

The Imaginary Unit, i Trivia 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!

A multiple-choice quiz by achernar. Estimated time: 7 mins.
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Author
achernar
Time
7 mins
Type
Multiple Choice
Quiz #
200,670
Updated
Jul 23 22
# Qns
15
Difficulty
Average
Avg Score
10 / 15
Plays
1509
Last 3 plays: Guest 75 (10/15), pollucci19 (5/15), WesleyCrusher (14/15).
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Question 1 of 15
1. How is the imaginary unit, i, defined? Hint


Question 2 of 15
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? Hint


Question 3 of 15
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?
Hint


Question 4 of 15
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".]
Hint


Question 5 of 15
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?
Hint


Question 6 of 15
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?
Hint


Question 7 of 15
7. Why, in electrical engineering and allied fields, is the imaginary unit denoted by the letter "j" (as opposed to "i")? Hint


Question 8 of 15
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?


Question 9 of 15
9. What result(s) does one get by taking the square root(s) of -36? Hint


Question 10 of 15
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?
Hint


Question 11 of 15
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? Hint


Question 12 of 15
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.]
Hint


Question 13 of 15
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)]?
Hint


Question 14 of 15
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])
Question 15 of 15
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? Hint



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quiz
Quiz Answer Key and Fun Facts
1. How is the imaginary unit, i, defined?

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.
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?

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.
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?

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.
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".]

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.
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?

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?

===
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
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?

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]

===
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.
7. Why, in electrical engineering and allied fields, is the imaginary unit denoted by the letter "j" (as opposed to "i")?

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!)
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?

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

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.
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?

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
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?

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.
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.]

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

===
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.
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)]?

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).

===
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).

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

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!

===
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.
Source: Author achernar

This quiz was reviewed by FunTrivia editor crisw before going online.
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