Consider the number 144

It’s a perfect square.

All its digits are perfect squares.

On reversing, it’s still a perfect square.

The sum of the digits is a perfect square

The product of the digits is a perfect square.

The sum of the digits is the square of the number of digits.

The square of the sum of the digits of the square root is the sum of the digits. “The square root of 144 is 12. The reverse of 144 is 441. The square root of 441 is 21 which is then again the reverse of 12, the square root of 144. Such numbers are called Adam’s numbers

 The difference between a million and a billion:

A million seconds is approximately 12 days while a billion is 31 years. Gabriel’s horn: An object with finite volume but infinite surface area, meaning you can fill it with paint but can’t paint it. It is constructed by rotating the part of the hyperbola y=1/x for x 

1 around the x-axis.

i^i = 0.20787957635

An imaginary number to the power of an imaginary number is a real number.

invention of chess

In primary school, I read this legend about the invention of chess, and it blew my mind. It goes like this:

Once, an emperor asked one of the most intelligent people in his empire to invent a new game for him. The man worked on it for days and finally presented the king with the game of chess.

“Name your reward,” the king said, delighted. The man said, “My wishes are simple. Give me one grain of rice for the first square, two for the second, four for the third, and so on for all the 64 squares, doubling the number of grains with each square.”

The king agreed, surprised that the man had asked for such a small reward. Later, the treasurer came and told the king that such an amount would add up to an astronomical number!

That was the first time I understood geometric progression or exponential growth. Here’s the breakdown of the problem: Most people would expect the total number of grains to be much lower (like the king did), but here’s how the number grows:

The total number of grains thus equals 18,446,744,073,709,551,615. That’s right, about 18 

Smallest Prime Number

When I was in my 12th, my sir asked: “Which is the smallest prime number?”

The whole class said that 2. My sir again asked ‘Why 1 is not the prime number?” At that moment the whole class was silent. Nobody in the class had the answer. I too was quiet. My sir started asking randomly “What is a prime number??” We said that the numbers which are divisible by 1 and itself. My sir said that we were wrong.

Then he told us the right definition “A natural number which is divisible by (±)1 and (±) itself having four factors is called a prime number.” Thus prime numbers have four factors.

Since 1 can be divided by only +1 and -1 so it has only two factors that’s why it can’t be considered as a prime number.

My sir told me that it was the question that was asked when he was doing his masters in maths and it was asked in his viva. I think that class will remain unforgettable for me for the rest of my life.

1 was considered a prime number until the beginning of the 20th century. Unique factorization was a driving force behind its changing of status since its formulation is quicker if 1 is not considered a prime, but I think that group theory was the other force. Indeed I prefer to describe numbers as primes, composites, and unities, that is numbers whose inverse exists (so if we take the set of integer numbers 

Measure Earth with a Scale and a Shadow

You can measure Earth with a scale and a shadow. It’s a hot summer day. The sun is beating down mercilessly. You’re walking down the road lost in your thoughts. Suddenly, you look down at the ground and notice that your shadow is missing!

You look up into the sky. The sun is right above your head. (Of course! That’s why there are no shadows). It’s the Summer Solstice. At your location, the rays from the sun are hitting perpendicular to the ground. Your shadow has vanished into a small area under your feet.

But the sun can’t be overhead everywhere on Earth at the same time, right? Two hundred miles from your home, there’s another city where the people aren’t so lucky. The sun appears a bit off-center in the sky. The shadows are still there.

But you’ve done something else too. Without even realizing it, you have also calculated another very important, very impossible-to-measure angle. The angle that the two cities subtend on the center of the planet!

You are AB. Sun’s rays are falling perpendicular on your head. If you extend this ray into the interior of the planet, as I’ve done, it would pass through the center of the Earth O.

Negative Numbers

When I was in the 4th grade my uncle, who has a PhD in Math, gave me a math book.

It taught me about negative numbers, which I thought were pretty cool.

It went on to discuss adding and subtracting numbers. One equation was something like:

$(+7)-(-3)=(?)$

Then it went on to give the rule that when you subtract a negative number, you just add its positive value:

$(+7)+(+3)=(+10)$

So I could do this but didn’t understand why the rule worked.

I am never satisfied when I don’t know why something works, so I asked my uncle. He said to determine this answer:

$(+7)+(-3)=(?)$

I knew this was like adding a debt of 3 dollars, so the equation is:

$(+7)+(-3)=(+4)$

Then he said to think of the original equation:

$(+7)-(-3)=(?)$

as removing a debt of 3 dollars. If the debt were removed, I’d be 3 dollars richer. So:

$(+7)-(-3)=(+7)+(+3)=(+10)$

And that made sense.

A Triangle on Top of a Rectangle,

When I was in grade 4 (or 5 be?), our math teacher drew one Figure/Diagram. There’s a triangle on top of a rectangle, in the following configuration.

Ask all students How much of the rectangle is covered Almost everyone got it wrong, with 2/3 a popular answer.

But when he drew a single vertical line you see it can be decomposed in two parts, each one half covered, everyone got it right:

The solution is thus exactly 50%. That day I learned the fact that one simple vertical line can solve the problem.

That is the power of Figures/Diagrams in Math.