Sunday, 11 February 2018

Maths in a minute: Compound interest and e

Submitted by Marianne on February 7, 2018
Spaghetti
Compound interest is a blessing for saving, but a curse for debt.
The only good thing about debt is that it's connected to one of the most important constants in maths: the number $e$.
To see the connection, suppose you borrow $\pounds 100$ at an annual interest rate of $3\% $. How much will you owe after three years if you don’t make any repayments? It’s tempting to think that you’ll owe the original $\pounds 100$ plus $9\% $ of $\pounds 100$, which is $\pounds 9$, giving a total of $\pounds 109$.
Unfortunately, though, this calculation is not quite correct. After one year you will owe the original amount, $\pounds 100$, plus the interest from the first year, which is
 \[ 0.03 \times 100 = \pounds 3. \]  
This gives a total debt of
 \[ 100 + 0.03 \times 100 = 100 \times (1+0.03) = \pounds 103. \]  
To find out what you owe after another year, you need to add the interest from the second year, which is
 \[ 0.03 \times (100 \times (1+ 0.03)) = \pounds 3.09. \]  
After the second year the interest is therefore
 \[ 100 \times (1+0.03) + 0.03 \times (100 \times (1+0.03)) = 100 \times (1 +0.03)^2 = \pounds 106.09. \]  
The interest in the third year will be
 \[ 0.03 \times (100 \times (1 + 0.03)^2) = \pounds 3.18, \]  
so after the third year you will owe a total of
 \[ 100 \times (1+0.03)^3 = \pounds 109.27. \]  
You can see that the amount of interest accrued in each year is more than the amount accrued in the previous year. If you’ve borrowed more than $\pounds 100$ this growth, or compounding, of the interest is much more noticeable: it is the curse of debt and, equally, the blessing of saving.
You may have seen a pattern in the expressions above. And indeed, there is a general expression describing how your total debt will grow:
 \[ P (1 + r)^ n \]  
where $P$ is the initial amount you borrowed, $r$ is the rate of interest (where $r$ is written as a decimal number, such as $0.03$, rather than a percentage, $3\% $) and $n$ is the number of times the interest is compounded. The more often the interest is compounded, the greater the total, which is where you have to be careful.
To make things simpler, suppose you borrow $\pounds 1$ ($P=1$) at an annual interest rate of $100\% $ ($r=1$). If the bank just calculates your total amount at the end of the first year, you will owe:
 \[ 1 \times (1+1)^1 = 1 \times 2 = £\pounds 2. \]  
But what if the bank decided to compound the interest quarterly, increasing the total every three months using an interest rate of $100/4 = 25\% $? Then after one year you would owe:
 \[ 1 \times ( 1+1/4)^4 = \pounds 2.44. \]  
Compounding monthly would result in a total of
 \[ 1 \times (1 + 1/12)^{12} = \pounds 2.61. \]  
Daily compounding would give
 \[ 1 \times (1+1/365)^{365} = \pounds 2.71 \]  
If the bank compounds $n$ times in a year you will owe:
 \[ 1 \times (1+1/n)^ n \]  
and the more times they compound, the greater the value for $n$, the more you will owe.
If you take things to the limit, calculating this for higher and higher values of $n$, you’ll find the total amount you’d owe converges to
 \[ 2.71828182845904523536028747135266249775724709369995... \]  
We call this number $e$, the limit of the value for our compounding formula as $n$ tends to infinity. The number $e$encapsulates growth, and can often be used to rewrite complicated equations describing growth in a much simpler way.
The person to first catch sight of the number $e$ in the context of compound interest was the mathematician Jacob Bernoulli in 1683. Bernoulli tried to find the limit of the expression
 \[ \left( 1+\frac{1}{n}\right)^ n \]  
as $n$ tends to infinity and managed to work out that it was between $2$ and $3$. It was the first time in history that a number was defined as a limit.
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Saturday, 10 February 2018

Indians Invented 'Zero' 500 Years Earlier Than Thought: Study

Researchers at University of Oxford in the UK found that the Bakhshali manuscript text contained hundreds of zeroes, putting the birth of 'zero' or 'nought' as it is also known, at 500 years earlier than scholars first thought.

Updated: September 17, 2017, 1:16 PM IST



Indians Invented 'Zero' 500 Years Earlier Than Thought: Study
A leaf from the Bakhshali Manuscript, showing off Indian mathematical genius. A zero symbol has been highlighted in the image (Photo: Bodleian Libraries Twitter)

London: An ancient Indian manuscript, dating back to the third century, has revealed the oldest recorded use of 'zero' - pushing back one of the greatest breakthroughs in the history of mathematics back by over 500 years, Oxford scientists say.

Bakhshali manuscript was found in 1881, buried in a field in what was then an Indian village called Bakhshali, now in Pakistan. It has been at the Bodleian Libraries in the UK since 1902.Researchers at University of Oxford in the UK used carbon dating to trace the origins of zero to the Bakhshali manuscript.

They found that the text contained hundreds of zeroes, putting the birth of 'zero' or 'nought' as it is also known, at 500 years earlier than scholars first thought.

The text dates back to the third or fourth century, making it the oldest recorded use of the symbol.
Previous studies asserted that the Bakhshali manuscript probably dated from between the 8th and the 12th century.

However, new carbon dating reveals that the reason why it was previously so difficult for scholars to pinpoint the Bakhshali manuscript's date is because the manuscript, which consists of 70 fragile leaves of birch bark, is in fact composed of material from at least three different periods.

"Determining the date of the Bakhshali manuscript is of vital importance to the history of mathematics and the study of early South Asian culture," said Richard Ovenden from Bodleian Libraries.

The concept of the symbol as we know and use today, began as a simple dot, which was widely used as a 'placeholder' to represent orders of magnitude in the ancient Indian numbers system for example 10s, 100s and 1000s, researchers said.

It features prominently in the Bakhshali manuscript, which is widely acknowledged as the oldest Indian mathematical text, they said.

The earliest recorded example of the use of zero was previously believed to be a 9th century inscription of the symbol on the wall of a temple in Gwalior, Madhya Pradesh.

Although a number of ancient cultures including the ancient Mayans and Babylonians also used the zero placeholder, the dot's use in the Bakhshali manuscript is the one that ultimately evolved into the symbol that we use today, researchers said.

"The creation of zero as a number in its own right, which evolved from the placeholder dot symbol found in the Bakhshali manuscript, was one of the greatest breakthroughs in the history of mathematics," said Marcus du Sautoy, Professor of Mathematics at the University of Oxford.

"We now know that it was as early as the 3rd century that mathematicians in India planted the seed of the idea that would later become so fundamental to the modern world. The findings show how vibrant mathematics have been in the Indian sub-continent for centuries," du Sautoy added.
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