Tag Archives: Mathematics

What are numbers and why do they do such a good job of explaining the universe?

For starters numbers are an abstraction. In reality, things just are. There is no number, because there are no category of things that can be repeated. No apple is truly the same as another and therefore a person cannot have more than one of anything. The real world is infinite in its complexity.

However, the human mind is not. The human mind is simple and must make assumptions and estimations to get along. The human mind considers an apple and another apple and doesn’t see their infinitely distinct reality. The mind sees an abstract simplified token – just an apple and another apple. Two apples.

This is a kind of magic. Representing several things as though it was a modified version of one thing, frees up the mind to do so much. It allows us to store large amounts of information outside of our bodies.

The simple human mind can only really conceive of about 3-6 things at once. If a person without counting is asked which group is larger and is shown two groups, one with 33 apples, and another with 31, is extremely difficult to tell. But with numbers a person can count. They can set aside the reality of the apples and use several kinds of abstract representation to tell how many there are. They can arrange the apples into groups of three – which can be easily identified – and use their fingers outstretched to represent their place in counting each group. This is storing information outside of oneself.

This is a profound transformation. It can be shown that numbers are a kind of representative logic. Adding the ability to store information outside the human body transforms humans from just an animal into turing complete. Turing machines can solve any problem that is computable given enough time.

To the extent that we are right that one thing is like another thing, abstraction and counting save us a lot of brainpower. It’s a kind of compression. When we use numbers to represent things, we discover that there are certain logical properties that can rearrange these groups (numbers) in ways that are more understandable without affecting their accuracy or changing the number at all. For instance, three groups of 10 apples is the same as 30 apples. Multiplying doesn’t do anything to the groups but it does make a simpler token to represent it in our memory (30 as opposed to 3 sets of 10).

These conceptual simplifications let us represent other relationships we discover. Like the fact that planets (from the Greek for wanderer) seem to look like stars that moves throughout the sky. By putting numbers on how much they move we can compare this that are hard to directly observe – just like the large groups of apples. And we can store that information outside of our minds so we can compare it over long periods of time.

Comparing these numbers lets us discover patterns that describe how the planets behave like Newton’s equations of motion and gravitation. What’s more, they let us predict how they will behave.

The most interesting and readable author on the subject is Bertrand Russell in his book Introduction to Mathematical Philosophy.

Why is 0 factorial = 1? If n = 0, shouldn’t the answer be 0 as anything multiplied by 0 = 0?

A factorial represents the number of ways you can organize n objects.
There is only one way to organize 1 object. (1! = 1)
There are two ways to organize 2 objects (e.g., AB or BA; 2! = 2)
There are 6 ways to organize 3 objects (e.g., ABC, ACB, BAC, BCA, CAB, CBA; 3! = 6).
How many ways are there to organize 0 objects? 1. Ergo 0! = 1.
This is consistent with the application of the gamma function, which extends the factorial concept to non-positive integers.

How is maximum occupancy of a building is calculated?

The calculation of maximum occupancy of a building has a significant place in the hazard management. The idea is increasingly influencing new construction methods.

The maximum occupancy of a building is calculated primarily based on two factors. One is the number of available exits in the building and the other is the use of space. The International Building Code has laid down some rules regarding the maximum occupancy of an area. IBC is a model building code developed by the International Code Council (ICC).
As per the definition given by IBC, an exit is a continuous and unobstructed path of vertical or horizontal egress travel from any occupied portion of the building or structure to a public way. Usually, the doors from kitchen and unused rooms are not considered exits.
When calculating the occupancy figure for a building, the two following calculations are used.
1. Floor space factor – The number of persons who can safely reside in the premises. Number of people = Floor area (m²) / Occupant density
2. Exit factor – The width and capacity of the exit routes to allow people to escape safely.
Whichever the less between these figures is the maximum occupancy of a building.
According to the building regulations, the occupant density varies depending on the nature of a building. As per the IBC recommendations, a standing/bar area should have an occupant density of 0.3 M²/person while a shop area could have 2 M²/person and an office area must have 6 M²/person.
Thus, for a bar with an area of 300 M², the maximum occupancy will be 1000. At the same time, an office space with the same area would have a maximum occupancy of 50 as per the floor space factor.
Now we consider the exit factor. As per the recommendations, the ideal width of an escape route or exit is 1050 mm. In any case, it should not be less than 750 mm. An exit with a width of 1050 mm can accommodate 200 Persons in normal conditions. An additional 15 Persons can be accommodated per every 75 mm. If the premises have multiple exits, the wider one is considered to be unavailable. Suppose the above mentioned bar has an exit of 1200 mm width and three exits 1050 mm wide. The total number of persons the exits can accommodate will be 600. Since it is the smaller figure, the maximum occupancy of the premises will be 600.

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What is the formula to calculate IQ? Is IQ of 100 good?

Intelligence is an abstract concept. It is difficult to measure intelligence level and express it in numbers. However, the term Intelligence Quotient is used for the purpose. This term became popular about a hundred years ago in 1916 when a Stanford University psychologist named Lewis Terman invented a formula to numerically measure the amount of intelligence in a human being. The formula was revised in 1937, then in 1960, and the last time in 1972 there were a few changes done to the formula. The formula used today to measure IQ is as follows:

IQ = Mental Age ÷ Physical Age x 100

In the formula, physical age is one’s age from the date of birth, but mental age is a completely different matter. To explain with an example, if a 10-year-old correctly answers questions which can normally only be answered by a 13-year-old, then even though his physical age is 10 years, his mental age is considered to be 13 years. Thus, according to the formula, his IQ would be 13 ÷ 10 x 100 = 130. On the other hand, if a 10-year-old answers questions which can normally only be answered by a 10-year-old (and he can’t answer tougher questions), then his IQ would be 10 ÷ 10 x 100 = 100. It means his mental and physical age are the same.

IQ Distribution

A person with IQ of 100 can not be said to have a sharp mind, because the IQ figure of 100 is normal. It also means that majority of the people the in world have an IQ of 100. Refer above the distribution of Intelligence Quotient.

To tell the truth, human intelligence level can not be determined mathematically with accuracy, intelligence being an abstract concept. Therefore, the formula for measuring IQ is not very reliable. According to this formula a woman named Marilyn vos Savant has IQ of 228, whereas Albert Einstein’s IQ is said to be 160.

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When was the mathematical symbol π (pi) discovered? And what are its peculiarities?

We know ‘pi’ as a constant used in geometry. It is the sixteenth letter of ancient Greek alphabet that is pronounced as ‘pie’. (FYI, ancient Greek alphabet consisted of 24 letters. The first letter being alpha and the last being omega. One may say ‘from alpha to omega’ instead of ‘A to Z’) When ‘pi’ is used in geometry it is no more a letter of alphabet but becomes a symbol denoting the ratio of the circumference of a circle to its diameter, which is approximately 3.14159256. In order to arrive at the value of the circumference of a circle one has to multiply its diameter with ‘pi’. It is as simple as that. π also helps in arriving at answers to complex problems without involving in-depth knowledge of mathematics or geometry.
Here is an example: If diameter of the Earth at the equator is exactly 12,756.274 kilometers, how many days would a car traveling at the constant speed of 40 kilometers per hour take to circumnavigate the Earth along the equator? Note that no information about the circumference is given in the data. Still one can find it as 40,075 kilometers by multiplying the diameter (12,756.274 kilometers) with π (3.14159256). Then by dividing the circumference by 40 one can find the time taken in hours and finally in days as 41 days 17 hours and 57 minutes. Just as the diameter of the Earth at the equator as well as the time required for circumnavigation could be found with such great ease the formula for finding the area of a circle viz. (πr²) is also not difficult.

The value of π was first calculated as 3.1416 by the great Greek mathematician Ptolemy. As the decimal system was not known in those days the value arrived at by Ptolemy was not accurate. Only after the decimal system became known in the 17th century it was learned that there was no end to the fractional digits after the whole number 3. An English mathematician named William Shanks spent 15 years of his life trying to fathom ‘pi’ till the end. He calculated up to the 707th place after the decimal. It was the year of 1874 when he calculated the 707th place but when his calculations were checked through the world’s first computer in 1945 a great error was detected. It was found that Shanks had made an error in calculating the 527th place after the decimal point and that made the values of all the subsequent 180 places wrong. Shanks had devotedly worked for 7 years over these incorrect calculations! Fortunately, the poor man was not alive in 1945 to hear the shocking news!

As the value of ‘pi’ is an irrational number having infinite number of figures after the decimal point any figure one decides to settle on can not be accurate. The longest calculation of the value of ‘pi’ has been made by the supercomputer Cray-2 in 1986. It calculated the value up to 2,63,60,000 places after the decimal point.

More reading:
Pi (Wikipedia)