Tag: math

Posted in Science & Nature

Pigeonhole Principle

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If I have three gloves, there must be at least either two left gloves or two right gloves. It is impossible to have one left glove, one right glove and a third glove that is neither left nor right (usually). This logic is called the pigeonhole principle. It is named because of the logic that if you have n pigeons and m pigeonholes where n > m (e.g. 10 pigeons in 9 holes), then at least one pigeonhole must contain more than one pigeon. This is because the biggest spread of the pigeons is putting at least one in each box, but as n > m, there is a pigeon left over and it must go in a box with another pigeon. The pigeonhole principle seems like a basic counting principle, but its implications are quite interesting.

For example, let’s say that your sock drawer is very unorganised and has a mix of black and white socks. What is the minimum number of socks you need to pick out before you get two of the same colour? The pigeonhole principle dictates that when n > m, each “slot” must be filled with more than one item. Here, the slot is colour. As there are two colours (m = 2), you only have to pick three socks out to have a matching pair (n = 3, 3 > 2).

The pigeonhole principle allows us to make seemingly impossible conjectures, such as the fact that a person living in London will have the exact number of hairs on their head as at least one other person living in London. An average human head has about 150,000 hairs and it would be a safe assumption to say that no one would have more than a million hairs on their head (m = 1,000,000). The population of London far exceeds a million (n > 1,000,000), therefore, there must at least two people living in London with the exact same amount of hair on their head. Similarly, if you are in a room with 366 other people, you are guaranteed to share a birthday with at least one person.

Posted in Science & Nature

Pi

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Pi (π) a mathematical constant that is defined as the ratio of a circle’s circumference to its diameter. It is approximately equal to 3.14159, but since it is an irrational number (cannot be expressed as a ratio), the decimal places go on and on with no repeating segments. The history of pi extends back to almost 5000 years ago, as it plays such a crucial role in geometry, such as finding the area of a circle (A = π ²). It is not an understatement to say that pi is among the top five most important numbers discovered in history (0, 1, i and e being the others).

The interesting thing about pi is that it is an irrational number. As mentioned above, this means that pi has an infinite number of non-repeating decimal places, with numbers appearing in random sequence. For example, pi to a 30 decimal places is 3.141592653589793238462643383279… Because of this feature, pi contains all possible sequences and combinations of numbers at a certain point. The corollary to this fact is, if pi is converted into binary code (a number system of only 0 and 1, used by computers to encode information), somewhere in that infinite string of digits is every combination of digits, letters and symbols imaginable. The name of every person you will ever love. The date, time and manner of your death. Answers to all the great questions of the universe. All of this is encoded in one letter: π.

That, is the power of infinity.

Posted in Science & Nature

Mathematical Beauty

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What is the most “beautiful” mathematical equation? For millenia, many mathematical formulas and concepts have been described as beautiful (and some defining beauty, as the golden ratio does). In the mathematical world, the adjective “beautiful” is used in the sense that certain mathematical concepts, despite the fact they are rational and objective, are so pure, simple and elegant that they can only be described as art.

One such formula is Euler’s identity:

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Renowned physicist Richard Feynman described it as “the most remarkable formula in mathematics”. What makes this array of symbols and numbers so beautiful? Firstly, it contains the three basic arithmetic operations exactly once each: addition, multiplication and exponentiation. It also connects five fundamental mathematical constants with nothing other than themselves and the arithmetic operations.

0 is the additive identity, as adding it to another number results in the original number. 1 is the multiplicative identity for the same reason as 0. Pi(π) is one of the most important mathematical constants in the history of mathematics that is ubiquitous in Euclidean geometry and trigonometry. Euler’s number(e) is the base of natural logarithms and is used widely in mathematical and scientific analysis. i(√-1) is the imaginary unit of complex numbers, a field of imaginary numbers that are not “real”, allowing for the calculation of all roots of polynomials. Euler’s identity neatly sums up the relation between these five numbers that are so crucial in the field of mathematics. It is also interesting to note that these five numbers were discovered at different points in history spanning over 3000 years.

Some people describe mathematics as a distinct language in itself. Not only that, but mathematics is considered the universal language as it is both universal and ubiquitous. If that is the case, than Euler’s identity can be considered an extremely pithy literary masterpiece.

Posted in Science & Nature

Folding Paper

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Take any piece of paper and fold it in half. Then fold it in half again. Chances are, you will not be able to fold the paper more than seven times. Try it. No matter how thin the piece of paper is, it is extremely difficult to fold a piece of paper in half more than seven times. The reason? Mathematics.

A standard sheet of office paper is less than 0.1mm thick. By folding it in half, the thickness doubles and becomes 0.2mm. Another fold increases it to 0.4mm. Already, the problem can be seen. Folding a paper in half doubles the thickness, meaning every fold increases the thickness exponentially (2ⁿ). By seven folds, the thickness is 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 times the original thickness. This makes the piece of paper so thick that it is “unfoldable”.

Another limitation is that folding the paper using the traditional method means the area also halves, decreasing exponentially. With a standard piece of paper, the area of the paper is so small after seven folds that it is mechanically impossible to fold it. Furthermore, the distortion caused by the folds is too great for you to apply enough leverage for folding the paper.

Could these limitations be overcome by using a larger piece of paper? Sadly, no matter how large the piece of paper, it is impossible (or at least extremely difficult) to fold a piece of paper over seven times. This has been a mathematical conundrum for ages, until it was solved in 2002 by a high school student named Britney Gallivan. Gallivan demonstrated that using maths, she could fold a piece of paper 12 times. The solution was not simple though. To fold the paper 12 times, she had to use a special, single piece of toilet paper 1200m in length. She calculated that instead of folding in half every other direction (the traditional way), the least volume of paper to get 12 folds would be to fold in the same direction using a very long sheet of paper.

Mathematics, along with science, is what makes something that seems so simple, impossible.

Posted in Life & Happiness

True Love

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Let us assume that everyone has a person they are destined to end up with. Can we calculate the probability of such a fateful meeting between a man and a woman?

Suppose that the woman is X and the man is Y. Firstly, X and Y need to be born as human beings. They cannot be born as a worm or an onion or something. Here, we will say that the total number of species is M and the population number of each species as P (technically this part is forcing it slightly, so we can skip it).

Although the two have to beat ridiculous odds just to start, just being born as human beings is not enough. One must be born with XX chromosomes to be a woman, and the other must be born with XY chromosomes to be a man.

Let us assume that the two were lucky enough to be born as a man and a woman. Next, they must live in the same space. If one lives in some Korean city and the other lives in some American rural village, it is unlikely the two will ever meet.

Even if they did live in the same place, X and Y must have subjective qualities that the other person finds attractive. If they are not interested in each other, nothing will happen even if they did meet. By this stage, we have clearly gone past the scopes of mathematics.

Then let us assume that a man and a woman, who fit each other perfectly and born as people, are living in the same space. We are still missing one variable: time. Even if we took only the 5000 years that civilisations have existed, the odds of the two being born in the same era as similar ages is less than 0.001%.

Species, sex, space, time… Statistically speaking, the chances of a man and a woman beating all of these odds to establish a perfect couple seem nearly impossible. But we can clearly see that “true love” exists all around us. Numbers are just numbers. If you find a person that makes your heart skip a beat when your eyes meet, that makes you feel that the more you get to know them, the more you think you cannot live without them; in essence a person that makes you think “this person is The One”, do not let the person slip away. The scenario of you and that person existing on the same space-time and loving each other is something that verges on the impossible.

There is no treasure as rare as true love. If you have found true love, or believe that you have found it, fight to seize it and do everything in your power to protect it. That is the greatest accomplishment you can make in life.

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Posted in Science & Nature

Seven Bridges Of Konigsberg

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The city of Königsberg (capital of Prussia, now Kaliningrad, Russia) has the Pregel River running through the middle, with islands at the centre of the river connected by seven bridges. Is it possible to cross all of these bridges while only crossing them only once each?

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If you try to solve this problem, you soon discover that it is incredibly difficult not to cross the same bridge twice. But it is difficult to tackle this problem in a brute force manner. To calculate all of the permutations in the order of bridges, you use 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040, meaning that there are 5040 possible arrangements of bridges. Then how can you prove if the problem is solvable or not?

The great mathematician Leonhard Euler, upon being asked to solve the problem, is reported to have said that the problem is impossible to solve on the spot. In 1735, he proved his answer by modelling the seven bridges of Königsberg in a diagram of four dots connected by lines (representing the bridges).

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By using this model, the problem is converted into a “draw in one stroke” problem, which is also called a Euler walk to honour Euler’s contributions. Euler discovered many properties and laws regarding such problems. If a certain point is the starting point, then the line must first leave the point, then even if it comes back to the point, it must leave again. Ergo, the starting point must have an odd number of lines connected to it. The opposite applies to the ending point, where a line must enter the point, and if it leaves the point it must come back to it. Ergo, the ending point must also have an odd number of lines connected to it. In the case of a Euler walk, the starting and ending points are identical, so the number of lines is the sum of two odd numbers, making it an even number. Thus, to find out whether a picture can be drawn using one line, use the following laws:

  1. If there are no points of odd degree (odd number of lines), the starting and ending points are identical.
  2. If there are two points of odd degree, the starting and ending points are different.
  3. If there are one of more than two points of odd degree, it is impossible to draw using one stroke.

Thus, a Euler walk is only possible if there are 0 or 2 points of odd degree. Looking at the seven bridges of Königsberg problem, we can see that A is connected to 5 lines and B, C and D are connected to 3 lines each. As there are four points of odd degree, we have thus proved that it is impossible to draw a path that crosses all the bridges while not crossing any bridge more than once.

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Posted in Science & Nature

Marriageable Age

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When is the right time to get married? According to Professor Tony Dooley, you can use an equation to find the right age for proposing. To do this, take “the youngest age you want to marry” and minus it from “the oldest age you want to marry” then times 0.368. Add this number to the youngest age. For example, if you would consider getting married from age 21 onwards and at the latest 30, your ideal age to marry is: (30 – 21) x 0.368 = 3.312 + 21 = 24.312, thus about 24 years and 4 months old. 

This equation is very practical as it is a modified version of equations used in financial and medical fields. This equation is used to maximise profit while minimising loss using mathematics. It may not sound romantic, but according to Professor Dooley, after you reach the calculated age you should not waste time and ask the hand of the next person you date in marriage.

(Source: http://soulofautumn87.deviantart.com/art/All-We-Need-Is-A-4-Letter-Word-111260511)

Posted in Science & Nature

Quadratic Formula

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Anyone who has studied mathematics to some degree will know about algebraic equations. An algebraic equation is an equation that can be solved to find the unknown value of x. A quadratic equation is an algebraic equation with x², or in other words has two valid solutions to x. Generally speaking, a quadratic equation can be expressed in the following fashion: ax² + bx + c = 0. a, b and c are constants and the equation can be solved to find x. A quadratic equation is definitely more complicated to solve compared to a linear equation and it can be solved using various means and applications such as factorisation. As these methods are learnt in school and this Encyclopaedia is technically not a mathematics textbook, such methods will not be delved into.

If you have not learnt it already, there is a shortcut method to solving quadratic equations: the quadratic formula. This formula can easily find x if you simply substitute in the values for a, b and c. Of course this formula only works if the solutions are real numbers. The quadratic formula is as follows:

As you can see, because of the ± sign, the formula can be used to find both solutions to a quadratic equation. Even without factorising, it can find the answer as long as you substitute numbers into it on a calculator, making maths class very easy. However, as mentioned above the Encyclopaedia of Absolute and Relative Knowledge is not a mathematics textbook and one should instead learn properly from their teacher, not using the formula until they have been taught it properly.

Posted in Science & Nature

Golden Ratio

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The golden ratio is a magical number that divides a line into the most beautiful ratio. It bestows a mystical power in an object and allows for the creation of excellent architecture and art.
This magical ratio is (1 + √5)/2, or 1.618033988. If there is a line divided by the golden ratio called a + b, then b:a and a:(a + b) are both the same ratio.

We can find the golden ratio in countless values seen in animals and plants. A snail shell’s golden spiral allows for the snail to grow without changing shape, while the distribution of branches on a tree also follows the ratio. The golden ratio controls everything from the spiral pattern of galaxies to the pattern of our brain waves. The golden ratio is the law of the universe.

Using this magical ratio, we can find the most beautiful composition of a human being. The Venus of Milo, considered as one of the most beautiful figures in history, has a ratio of 1:1.618 between her upper and lower body (divided at the belly button) – the golden ratio. The same can be said for the ratio between the head and neck compared to the rest of the upper body, and the length from the belly button to the knee compared to the length below the knee. The exact same composition was used to construct the statue of Doryphoros, one of the most famous examples of ancient Greek sculptures. The diagram that illustrates these ratios is the Vitruvian Man by Leonardo da Vinci (Vitruvius was a Roman architect who utilised the ancient Greek knowledge of applying the proportions of a human being, i.e. the golden ratio, in constructing temples). 

The Great Pyramids of Giza, Solomon’s Temple and the Parthenon are all partially constructed according to the golden ratio. It is said that buildings constructed outside of the golden ratio will collapse over time. The same is seen in Eastern constructions, such as buildings and inventions from the Goryeo Dynasty of Korea. 

Interestingly, the golden ratio applies to intangible objects as well. For example, Chopin’s Nocturne pieces tend to climax at the point of the golden ratio (roughly two-thirds in). The ratio is still used in modern day design, with the standard credit card size being the best example.

The golden ratio is an eternal beauty that does not go out of fashion with time.

Posted in Science & Nature

Euclidean Geometry

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In 300BC, a Greek mathematician called Euclid wrote a series of texts called Elements. The Elements was a textbook that outlined many principles of mathematics (especially geometry) and it would become one of the most influential works in the history of mathematics. It is composed of a series of axioms (the axiomatic approach) from which many deductions and theorems can be made. Although many of these axioms sound extremely simple and like common sense, the implications are staggering.

The following is Euclid’s Five Postulates of Plane Geometry:

  1. Two points determine a line. 
  2. Any line segment can be extended in a straight line as far as desired, in either direction. 
  3. Given any length and any point, a circle can be drawn having the length as radius and that point as centre. 
  4. All right angles are congruent (can be superimposed). 
  5. Parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side, if extended far enough. 

Using these postulates, mathematicians are able to deduce more advanced theories. For example, the Elements also describes the famous Pythagorean theorem, which states that “in any right triangle, the area of the square of the hypotenuse (the diagonal) is equal to the sum of the areas of the squares of the other two sides” (a² + b² = c²).

Thanks to Euclid’s works, we are now able to accurately model and measure the three-dimensional space around us. Not only did Euclid set the foundations for mathematics, his works were also instrumental in the development of logic and modern science.