Among the lesser known ( at least to a high school student) mathematical constants that exist in the world rests Euler's number. This number is different from most other mathematical constants in that it cannot be represented in a geometric fashion: that is to say, by a shape.
Take the Pythagorean and Theodorus constants, for example (the square roots of 2 and 3 respectively), which can be represented by right angled triangles. The Pythagorean constant is the hypotenuse of a right triangle whose other two sides are of a unit length, and the Theodorus constant is the hypotenuse of a right triangle having a side of length 1 unit and another of length square root of 2 units. Pi is also a popular example: it is the circumference of a semicircle having a radius of unit length.
Unlike these three, Euler's number cannot be represented by a shape: it can only be represented by a branch of math we call calculus. The reason for this is that calculus is the mathematics of growth, and as I will shortly show you, Euler's number is also heavily related to growth.
Say you have a rupee which you want to deposit in a bank. The bank, in a show of generosity, agrees to give you 100% interest on the rupee per annum. Therefore, at the end of every year, you get two rupees, instead of one, in your bank account. However, if the bank decides to offer 50% interest every six months instead, would that be a better deal?
The answer: indeed, it would be a better deal. At the end of the first six months, your bank account would have 1.5 rupees in it, and at the end of the second, it would have 1.5 + 50% of 1.5; that is, 2.25 rupees. So, you end up with 25 paise extra! Similarly, if you earned 25% interest every 3 months, you would earn approximately 2.44 rupees, earning 44 paise extra. Long story short: you get more money by reducing the percent interest on your savings in a bank account and increasing frequency of interest gained.
Thinking in this vein, what would be the value of the final amount in the bank account if one earned interest every instant: every second increasing the money in one's account? Well, that's also what Jacob Bernoulli questioned, and tried to figure out, but in the end, it was Leonhard Euler who figured out the answer:
2.71828183.........
To figure out this number, one traditionally uses the formula given below:
but Euler used a different formula, namely
1 + (1/1!) + (1/2!) + (1/3!) + (1/4!) +..... ad infinitum
and worked out the number up to eighteen decimal places. So, as is immediately visible, Euler's number is heavily related to growth, and is heavily used in calculus as a result.
This constant can also be represented in a graphical manner, with the function y = e^x, with the y-intercept of the graph being 1 (as if x is 0, y = e^0, which is always 1). It can be ascertained that if x=1, y becomes e, making the chief coordinates of the graph (0,1) and (1, 2.718...). The reason this graph is so special is that the slope of the curve, the area underneath it and the y-coordinate of a point along the curve are all equal. This unique property makes e the natural language of calculus, and using it in calculus makes the math much simpler.
Lastly, if we are talking about Euler's number, we have to make mention of the most beautiful equation in mathematics:
Seems complicated, doesn't it? This equation is littered with constants, including e itself, pi, i (meaning iota, the square root of -1), 1, and 0. It has been proven using concepts of calculus, complex numbers, trigonometric functions and the concept of unit circle. By fiddling with the equation for a bit, it becomes clear that:
as i is the root of -1.
Take the Pythagorean and Theodorus constants, for example (the square roots of 2 and 3 respectively), which can be represented by right angled triangles. The Pythagorean constant is the hypotenuse of a right triangle whose other two sides are of a unit length, and the Theodorus constant is the hypotenuse of a right triangle having a side of length 1 unit and another of length square root of 2 units. Pi is also a popular example: it is the circumference of a semicircle having a radius of unit length.
Unlike these three, Euler's number cannot be represented by a shape: it can only be represented by a branch of math we call calculus. The reason for this is that calculus is the mathematics of growth, and as I will shortly show you, Euler's number is also heavily related to growth.
Say you have a rupee which you want to deposit in a bank. The bank, in a show of generosity, agrees to give you 100% interest on the rupee per annum. Therefore, at the end of every year, you get two rupees, instead of one, in your bank account. However, if the bank decides to offer 50% interest every six months instead, would that be a better deal?
The answer: indeed, it would be a better deal. At the end of the first six months, your bank account would have 1.5 rupees in it, and at the end of the second, it would have 1.5 + 50% of 1.5; that is, 2.25 rupees. So, you end up with 25 paise extra! Similarly, if you earned 25% interest every 3 months, you would earn approximately 2.44 rupees, earning 44 paise extra. Long story short: you get more money by reducing the percent interest on your savings in a bank account and increasing frequency of interest gained.
Thinking in this vein, what would be the value of the final amount in the bank account if one earned interest every instant: every second increasing the money in one's account? Well, that's also what Jacob Bernoulli questioned, and tried to figure out, but in the end, it was Leonhard Euler who figured out the answer:
2.71828183.........
To figure out this number, one traditionally uses the formula given below:
1 + (1/1!) + (1/2!) + (1/3!) + (1/4!) +..... ad infinitum
and worked out the number up to eighteen decimal places. So, as is immediately visible, Euler's number is heavily related to growth, and is heavily used in calculus as a result.
This constant can also be represented in a graphical manner, with the function y = e^x, with the y-intercept of the graph being 1 (as if x is 0, y = e^0, which is always 1). It can be ascertained that if x=1, y becomes e, making the chief coordinates of the graph (0,1) and (1, 2.718...). The reason this graph is so special is that the slope of the curve, the area underneath it and the y-coordinate of a point along the curve are all equal. This unique property makes e the natural language of calculus, and using it in calculus makes the math much simpler.
Lastly, if we are talking about Euler's number, we have to make mention of the most beautiful equation in mathematics:
Seems complicated, doesn't it? This equation is littered with constants, including e itself, pi, i (meaning iota, the square root of -1), 1, and 0. It has been proven using concepts of calculus, complex numbers, trigonometric functions and the concept of unit circle. By fiddling with the equation for a bit, it becomes clear that:
as i is the root of -1.
Very interesting blog. Though we have used e in calculus, certain things I knew only after reading the blog. Very good work Ishan
ReplyDeleteThanks so much for your words aaba.
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