The Square Root of 2 - That's Irrational! Online Quiz | Sci / Tech

Quiz Answer Key and Fun Facts

1. The approximation of the square root of 2 to five decimal places is given by which of the following numbers?

Answer: 1.41421...

A clever mathematical mnemonic that is associated with the first few digits of the square root of 2 is given by the sentence "I have a root of a two whose square is two", which corresponds to the number 1.4142135623.

The three wrong options are pi (3.14159...), phi or the golden ration (1.61803), and Euler's number (2.71828...); all of which are irrational mathematical constants.

2. For practical purposes, which of the following fractions is being used as a rational approximation to the square root of 2?

Answer: 99/70

Knowing that the square root of 2 has a value greater than 1, this eliminates the two options 70/99 and 7/22. The fraction 22/7 is a rational approximation used for pi (3.142...).

The fraction 99/70 yields 1.4142857143..., which is very close to the actual value of the square root of 2, which is 1.4142135623...

Though other fractions might give even better approximation, 99/70 is preferred by engineers and mathematicians as both the numerator and denominator of this fraction are reasonably small numbers.

3. It is known that the square root of 2 is a solution of the polynomial equation displayed in the photo. This being said, the square root of 2 can be classified as which of the following numbers?

Answer: Algebraic number

An algebraic number refers to any number that is a root (solution) of a non-zero polynomial equation with integer coefficients. The photo shows a quadratic equation, where the coefficients of x^2, x, and constant are 1, 0, and -2, respectively. The two roots of the quadratic equation are given by the square root of 2 (1.412...) and the negative square root of 2 (-1.412...).

A perfect number is a positive integer that can be expressed as the sum of all of its positive divisors, except the number itself. The first few perfect numbers are 6 (1+2+3), 28 (1+2+4+7+14), and 496 (1+2+4+8+16+31+62+124+248).

Fibonacci numbers are generated from Fibonacci sequence by adding up the values of two preceding numbers. Starting from 0 and 1, the first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, and so on.

An imaginary number b*i is defined by a real number b, which is multiplied by the imaginary unit, i (the square root of -1).

4. The photo depicts an isosceles right triangle whose hypothenuse is given by the square root of 2. What is the length of the two shorter sides (denoted by the question mark in the photo)?

Answer: 1

Let's denote the length of the shorter side as c. Then, using the Pythagorean theorem, c^2 + c^2 = (square root of 2)^2. Simplifying the equation, we have 2c^2 = 2. Solving for c, we have c = -1, 1. We omit -1 because the length must take positive values. Hence, c = 1.

5. Which Greek philosopher, a follower of Pythagoras, is credited with the discovery of the irrationality of the square root of 2?

Answer: Hippasus

The discovery of the irrationality of the square root of 2 by Hippasus of Metapontum shocked and angered the Pythagoreans, as the Pythagoreanism doctrine believed that all numbers are rational.

Hippocrates of Kos was a Greek physician; he was known as the "Father of Medicine". The hippocampus is a structure of the brain. The hippopotamus is a large mammal.

6. Greek mathematician Euclid provided an elegant proof on the irrationality of the square root of 2. The proof starts with the assumption that the number is rational and can be expressed in the form of the fraction a/b, where both positive integers a and b share only a common divisor of 1. After some algebraic manipulation and simplification, it turns out that the greatest common divisor of a and b is NOT 1. What is the name given to this method of proof?

Answer: Proof by contradiction

As the name suggests, proof by contradiction, also known as reductio ad absurdum (by reduction to the absurd) in Latin, first assumes a statement which will later be proven to be invalid, which results in a contradiction.

The elegant proof provided by Euclid first assumes that the square root of 2 is indeed rational. If this is true, then the number can be expressed as a fraction in its simplest form, meaning that the fraction a/b can no longer be simplified. After some steps, it can be shown that both a and b are multiples of 2, which implies that the fraction is not written in its simplest form. This leads to a contradiction, which means that the original assumption that we began with is wrong. Therefore, we can conclude that the square root of 2 is irrational.

7. The photo depicts an elegant property of the square root of 2. In words, the reciprocal of the square root of 2 plus 1, is given by the square root of 2 minus 1. Which algebraic formula is associated with this property?

Answer: (a+b)(a-b) = a^2 - b^2

Let us denote a to be the square root of 2 (sqrt(2)) and b to be 1. Hence, (a+b)(a-b) = a^2 - b^2 = (sqrt(2))^2 - 1^2 = 2 - 1 = 1. Rearranging, we have 1/(sqrt(2)+1) = sqrt(2) - 1.

8. The reciprocal of the square root of 2 is a ubiquitous mathematical constant in the fields of geometry and trigonometry. Which value of theta satisfies the mathematical equation depicted in the photo?

Answer: 45

The reciprocal, or the multiplicative inverse of the square root of 2, approximates 0.7071067811...This value is also half the value of the square root of 2. Both sin 45 degrees and cos 45 degrees can be expressed in their exact forms of 1/sqrt(2).

9. The square root of 2 can be found in many fields. The value gives the frequency ratio of a tritone interval. In which field of study would one encounter a tritone interval?

Answer: Music

In music theory, the most commonly-used tuning system is termed twelve-tone equal temperament, where it divides an octave equally into 12 parts. On a logarithmic scale, the ratio of two successive notes is given by the 12th root of 2. A tritone consists of six semitones, which gives rise to a frequency ratio of the square root 2.

10. The dimension of this ubiquitous item is given by 297mm x 210mm (11.7 in x 8.3 in). The length-to-width ratio approximates the square root of 2. What item is being described?

Answer: A4 paper

All papers of the "A" series, have the same length-to-width ratio that approximates the square roots of 2. An A4 paper measures 297 mm (11.7 in) in length and 210mm (8.3 in) in width. If an A4 paper is folded in half, the resulting paper's dimension is that of A5's, with 210 mm (8.3 in) as its length and 148 mm (5.8 in) as its width.

Notice that the length of an A5 paper matches the width of an A4 paper. The Lichtenberg's ratio (1.414...) was named after German physicist Georg Christoph Lichtenberg (1742 - 1799).

A typical name card of size A7 measures 105mm (4.1 in) in length and 74 mm (2.9 in) in width.

Source: Author Matthew_07

This quiz was reviewed by FunTrivia editor rossian before going online.
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The Square Root of 2 - That's Irrational! Quiz

The square root of 2, though not as famous as other mathematical constants such as pi and Euler's number, is notably associated with the various elegant proofs demonstrating the number's irrationality. Let's learn more about this special number. Enjoy!

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