Tomorrow, March 14, is 3/14 [in month/year form] and so we celebrate Pi Day! And since I’m teaching History of Math for the first time in rather a lot of years, I’m thinking that the perfect topic is the history of the symbol.
But first, what is 𝝅? The idea is that no matter how big or small a circle, the circumference is always a little more than 3 times as large as the diameter: that ratio is about 3.14 in decimal terms. Because it’s a ratio, the first symbols were also written as ratios: William Oughtred called it 𝝅/δ in 1647, and while he didn’t explain what either of the terms meant, since 𝝅 is the Greek p it likely stood for periphery (according to my source, though I think of perimeter myself when I see it); likewise, δ is the Greek d and likely stood for diameter. Several other mathematicians adopted this notation.
The first person to use a single symbol to represent this ratio was Johann Christoph Sturm, who in 1689 referred to it with the letter e. Wait, what? (Double check.) Well that is something I didn’t know before. Cool! But using e didn’t catch on, and less than twenty years later, in 1706, William Jones used the symbol 𝝅 for this same ratio. No explanation as to why, and also no consistency – he used the same symbol to mean other things earlier in the same book. This use of 𝝅 also didn’t catch on: other mathematicians continued to use other symbols for the ratio of a circumference to the diameter, and 𝝅 itself continued to be used for different mathematical numbers. But eventually, over the 1700s, its use caught on and so we have the well known symbol today.
Source: A History of Mathematical Notations by Florian Cajori
=n%5E2%2Bn%2B41)
. At first glance this seems to work:=0%5E2%2B0%2B41=41)
, which is prime.=1%5E2%2B1%2B41=43)
, which is prime.=2%5E2%2B2%2B41=47)
, which is prime.=39%5E2%2B39%2B41=1601)
, which is prime. =40%5E2%2B40%2B41=1681=41%5E2)
, which is not prime. 
is prime if and only if )
then 
and sure enough 
is a prime number.
, then 
and 
is also prime! 
then 
and 
is not prime.
this formula produces 
, which is the first prime. When 
it gives 
, which is the second prime. And so on – you get all the primes this way!!!! Pretty amazing. You can watch a youtube video all about it here:
: neverending.” But then you might get distracted thinking, well, technically all numbers are neverending. Even a number like 1 could be written as 1.000000…. So you’d try to pick a different property of 
. You could do something similar for any integer.
. You could do something similar for any fraction.
is algebraic because it’s a solution to 
. Oooh, and now things get interesting. Because 
, because it turns out that the set of algebraic numbers is closed under addition, subtraction, multiplication, and even division (as long as you don’t divide by 0). That means that if you add, subtract, multiply, or divide algebraic numbers, you get another algebraic number. It’s also closed under square roots, cube roots, etc. This means if you write a number like ![\sqrt{\frac43-\sqrt{\frac{2+\sqrt[5]{3}}{4+\sqrt{7}}}}](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Csqrt{%5Cfrac43-%5Csqrt{%5Cfrac{2%2B%5Csqrt[5]{3}}{4%2B%5Csqrt{7}}}})
it’s going to be an algebraic number and you don’t have to figure out what polynomial it’s a root of (although you can if you want).
is an algebraic number because it’s a root of 
, but even if you stick with real numbers there are algebraic numbers that can’t be written with the symbols described above. For example, the polynomial 
has a root between 1 and 2 (since 
is negative, but 
is positive), and that root will automatically be algebraic, but it turns out that root can’t be written in a closed form, meaning it can’t be written just with +,-,x,/, and roots. Isn’t that wild? 
is also transcendental. In fact, most numbers are transcendental, in the sense that the set of algebraic numbers is countable but the set of transcendental numbers is not. But it can be really hard to tell if a particular number is transcendental: we still don’t know, for example, if 
is algebraic or transcendental. Then again, we don’t even know if %5E{53})
(which is approximately 2,151,972,563.22) is very close to 
(which is 2,147,483,648). If you want to avoid fractions, this means that 
is almost the same as 
. They won’t be exactly the same, but it’s pretty close.%5E{12})
(which is approximately 129.7) is very close to 
, which is 128. 
. Or 
. (There are examples all over the internet, so a quick search reveals many solutions should you wish.)
(Heh heh — I just ran that page through Adobe’s Optical Character Recognition, and it pretty much didn’t recognize a thing.)
(or, more generally, as 
– in the previous equation I used 
).
. Here’s a short video of how it is constructed
360 
#/media/File:Diagram_of_I_Ching_hexagrams_owned_by_Gottfried_Wilhelm_Leibniz,_1701.jpg){kind=link}
