FREE! Click here to Join FunTrivia. Thousands of games, quizzes, and lots more!  # Complex Numbers: Real and Imaginary! Quiz

### Complex numbers are very useful in the fields of maths and physics and this is, hopefully, an interesting look at them in all their glory...even if only briefly! Good luck. :)

A multiple-choice quiz by jonnowales. Estimated time: 5 mins.

Author
jonnowales
Time
5 mins
Type
Multiple Choice
Quiz #
324,689
Updated
Dec 03 21
# Qns
10
Difficulty
Average
Avg Score
7 / 10
Plays
653
Awards
Top 35% Quiz
Last 3 plays: AdamM7 (10/10), Guest 222 (7/10), Guest 192 (8/10).
1. Complex numbers come in two parts and are seen in the form 'a + bi' where 'a' and 'b' are constants and 'i' is given the dreamy name, the imaginary unit. Which of those letters represents the *real* part of the complex number? Hint

None of them, all parts of a complex number are imaginary
b
a
i

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2. It is just a single letter that represents the difference between the real numbers and the complex numbers, and that letter is 'i'. What mathematical value does the letter 'i' represent? Hint

1^(1/2)
(-1)^(1/2)
(-1)^2
1^2

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3. All the main arithmetic functions (addition, subtraction, multiplication and division) can be applied to complex numbers of the form 'a + bi'. So, if there are two complex numbers, 'Z1 = a + bi' and 'Z2 = c + di', what is the value of 'Z1 + Z2'?

Clue: Add the real terms and imaginary terms separately.
Hint

(a + d) + (b + c)i
(a + c) + (b + d)i
(a + b) + (c + d)i
(a + a) + (b + b) + (c + c) + (d + d)i

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4. The division of a complex number by another complex number requires a little bit of algebraic manipulation before the arithmetic takes place. If there are two complex numbers, 'Z1 = a + bi' and 'Z2 = c + di', what must be found if 'Z1' is to be divided by 'Z2'? Hint

Complex conjugate of Z2 (Z2*)
Z1 + Z2
Complex conjugate of Z1 (Z1*)
Z1 - Z2

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5. Complex numbers don't just exist for the purpose of trivial addition and subtraction. There are many elegant formulae which involve complex numbers and one such formula is:

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

Who is this formula named for?
Hint

Carl Gauss
Abraham de Moivre
Leonhard Euler
Ren� Descartes

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6. The imaginary unit, 'i', makes an appearance in the famous mathematical formula:

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

This wonderful, elegant formula links 'e^(ix)' to a unit circle and is useful in simplifying the analysis of differential equations. Who of the following is this formula named for?
Hint

Ren� Descartes
Carl Gauss
Leonhard Euler
Abraham de Moivre

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7. Every now and again one hears a mathematician talk of the beauty of their subject and one particular identity is often heralded as the most beautiful of all. This identity is derived from the formula, 'e^(ix) = cos(x) + sin(x)i', and involved pi, 'e', and the imaginary unit, 'i'. After which of the following was the identity '[e^(i*pi)] + 1 = 0' named? Hint

Abraham de Moivre
Ren� Descartes
Carl Gauss
Leonhard Euler

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8. Complex numbers can be plotted on a two-dimensional plane, known simply as the complex plane. The two dimensions are in the x-direction and in the y-direction. Does the imaginary part of a complex number dictate how far along the x-axis a plot point is?

Yes
No

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9. Complex numbers are often considered vectors due to their similarities, particularly in arithmetic. Just as is the case with vectors, what is another name for the absolute value of a complex number? Hint

Argument
Modulus
Phase
Integrand

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10. It seems quite extraordinary when a word such as "imaginary" is heard when considering a number, that such a number has real-life applications in science! Which of the following analyses uses complex numbers in its dealings with waves and signals? Hint

Newtonian analysis
Fourier analysis
Planck's analysis
Riemann's analysis

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Most Recent Scores
Sep 23 2023 : AdamM7: 10/10
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Score Distribution Quiz Answer Key and Fun Facts
1. Complex numbers come in two parts and are seen in the form 'a + bi' where 'a' and 'b' are constants and 'i' is given the dreamy name, the imaginary unit. Which of those letters represents the *real* part of the complex number?

The best way to look at a complex number ('a + bi') is in two parts, the first part is 'a' and you'll notice it is not multiplied by the imaginary unit, 'i'. It, 'a', is therefore not imaginary and is thus real, however, 'b' is multiplied by the imaginary unit and so 'b' is the imaginary part of the complex number.

a = Real part

b = Imaginary part

i = Imaginary unit
2. It is just a single letter that represents the difference between the real numbers and the complex numbers, and that letter is 'i'. What mathematical value does the letter 'i' represent?

The imaginary unit is easier to define in words by first playing around with the algebra:

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

As can be seen above, the imaginary unit squared is equal to negative one. When students first start off studying mathematics and come across the quadratic equation (which features a square root), they are told that you can't take the square root of a negative number. Later on in one's studies however, it is found out that you can indeed square root a negative number, and 'i' is the result!
3. All the main arithmetic functions (addition, subtraction, multiplication and division) can be applied to complex numbers of the form 'a + bi'. So, if there are two complex numbers, 'Z1 = a + bi' and 'Z2 = c + di', what is the value of 'Z1 + Z2'? Clue: Add the real terms and imaginary terms separately.

Answer: (a + c) + (b + d)i

The addition of complex numbers is nice in the fact that it is very much similar to the addition of real numbers. The only extra process is to add the imaginary terms of complex numbers up as well. In 'Z1' the real part of the complex number is 'a' and in 'Z2' the real part is 'c'. The addition of these real parts is 'a + c', but you then have to add the imaginary terms which is basically the same. The imaginary term in 'Z1' is 'b' and in 'Z2' it is 'd' and the addition of these two terms is 'b + d'. The only thing left to do is bolt on the imaginary unit, 'i':

Z1 + Z2 = (a + c) + (b + d)i
4. The division of a complex number by another complex number requires a little bit of algebraic manipulation before the arithmetic takes place. If there are two complex numbers, 'Z1 = a + bi' and 'Z2 = c + di', what must be found if 'Z1' is to be divided by 'Z2'?

Answer: Complex conjugate of Z2 (Z2*)

In order to divide one complex number by another, the complex conjugate of the denominator needs to be discovered. If 'Z1' is to be divided by 'Z2' then 'Z2' is the complex denominator ('Z1' will be the numerator). We have defined 'Z2' as being 'c + di' and the complex conjugate of this is found by simply changing the sign associated with the imaginary part of the complex number.

Z2 = c + di

Z2* = c - di = complex conjugate
5. Complex numbers don't just exist for the purpose of trivial addition and subtraction. There are many elegant formulae which involve complex numbers and one such formula is: [cos(x) + sin(x)i]^n = cos(nx) + sin(nx)i Who is this formula named for?

Below is de Moivre's Formula:

[cos(x) + sin(x)i]^n = cos(nx) + sin(nx)i; where 'cos' and 'sin' are trigonometric functions, 'x' is an angle in radians, 'n' is a constant number and 'i' is the imaginary unit equal to the square root of negative one.

The enigmatic 'i' appears in this formula which means that complex numbers aren't far away. In fact if you arbitrarily pick an angle 'x' in radians and a random number 'n', the right hand side of the equation can be nicely reduced to the simplest form of a complex number, 'a + bi'.

Further inspection of the right hand side of de Moivre's formula leads one to notice that both 'cos(nx)' and 'sin(nx)' are examples of multiple angles. That is, the argument of both cosine and sine (the terms in the brackets) is a multiple, 'n', of the angle 'x'. If 'n = 2' then we get 'cos(2x)' and 'sin(2x)' and these are examples of double angles. The formula proposed by de Moivre can help quantify such multiple angles in terms of single angles, such as 'cos(x)'.
6. The imaginary unit, 'i', makes an appearance in the famous mathematical formula: e^(ix) = cos(x) + sin(x)i This wonderful, elegant formula links 'e^(ix)' to a unit circle and is useful in simplifying the analysis of differential equations. Who of the following is this formula named for?

Euler's Formula, 'e^(ix) = cos(x) + sin(x)i' is perhaps the most celebrated work of genius in the field of mathematics. Physicist and science icon, Richard Feynman, said of the formula, "[it is] one of the most remarkable, almost astounding, formulas in all of mathematics".

Euler's Formula is intricately linked with geometry and trigonometry; the former in the respect of the complex plane and Argand diagrams and the latter with regards to the derivation of hyperbolic functions.
7. Every now and again one hears a mathematician talk of the beauty of their subject and one particular identity is often heralded as the most beautiful of all. This identity is derived from the formula, 'e^(ix) = cos(x) + sin(x)i', and involved pi, 'e', and the imaginary unit, 'i'. After which of the following was the identity '[e^(i*pi)] + 1 = 0' named?

Euler's identity, '[e^(i*pi)] + 1 = 0'*, really is extraordinary. Even if the aesthetics of the identity don't tickle your fancy, the truly beautiful and inspiring aspect of Euler's identity is hiding beneath the surface.

The irrational number 'e' is named Euler's number and is the inverse of the natural logarithm, 'ln'. The value of 'e' cannot be accurately expressed in numbers zero to nine because there is no end; it struts off to an infinite number of decimal places. To illustrate the point I wish to make however, 'e = 2.71828182845904523536' to twenty decimal places (but there are infinitely many more).

Pi is a well known irrational number, which also has infinite decimal place values, and to twenty decimal places it is given the value, 'pi = 3.14159265358979323846'. The imaginary unit, 'i', is equal to the square root of negative one. The beauty is in the fact that the numbers 'e' and pi which both go on for an infinite number of decimal places can be placed into an expression as simple as '[e^(i*pi)] + 1' and equate to zero! The mathematics behind how these two phenomenal numbers end up interacting to equal zero is available but I'll leave it as an exciting mystery here!

*I'd encourage anybody, if they have not seen the expression before, to have a look around the internet or in mathematical books for this identity as it is more aesthetically pleasing in those formats.
8. Complex numbers can be plotted on a two-dimensional plane, known simply as the complex plane. The two dimensions are in the x-direction and in the y-direction. Does the imaginary part of a complex number dictate how far along the x-axis a plot point is?

The value of the real part of a complex number dictates how far along the x-axis (abscissa) the plot point is. The value of the imaginary part of a complex number corresponds to where the plot point is positioned with regards the y-axis (ordinate). The complex plane forms the basis of an Argand diagram and is thus sometimes referred to as the Argand plane.
9. Complex numbers are often considered vectors due to their similarities, particularly in arithmetic. Just as is the case with vectors, what is another name for the absolute value of a complex number?

The modulus of a complex number 'z', '|z|', is another way of saying the absolute value of 'z'. If 'z = a + bi', then '|z| = ([a^2] + [b^2])^1/2'. That is, the absolute value of a complex number is found by squaring the value of the real part, squaring the value of the imaginary part, adding the squares together and finally taking the square root of that value. You may notice that this is just applying Pythagorean theorem to complex numbers, 'z^2 = a^2 + b^2'.
10. It seems quite extraordinary when a word such as "imaginary" is heard when considering a number, that such a number has real-life applications in science! Which of the following analyses uses complex numbers in its dealings with waves and signals?

One of the most basic uses of complex numbers in the physics of waves is best demonstrated with the use of a sine wave with a given frequency. By finding the modulus, or absolute value, of a complex number 'z = a + bi', '|z|', you have simultaneously found the amplitude of the sinusoidal wave. By finding the argument of 'z' (inverse tangent of the quotient of 'b/a') you find the phase of the wave.
Source: Author jonnowales

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