It's Presidents' Day and Cadence is on holiday. So time for me to write about something that is interesting (well, it's interesting to me, anyway) but not directly anything to do with semiconductors or EDA. The Fundamental Theorem of Arithmetic Somebody asked me recently why 1 is not a prime number. After all, it is only divisible by itself and 1, which is the most concise definition. In fact, a prime number is any number greater than 1 that is only divisible by itself and 1, and so the definition explicitly excludes 1. We (by which I mean mathematicians through the years) defined what a prime number is, and we could have made 1 a prime number without it being a silly definition (which it would be if we defined a prime number to be any number containing a 7, for instance—there wouldn't be much you could use that concept for). If 1 was a prime number, then some things in math wouldn't require an exception. For example, Goldbach's conjecture states that any even number greater than 2 is the sum of two primes. The reason for the "greater than 2" wrinkle is because 2 is 1+1 and we have defined 1 not to be prime. If 1 was prime, then Goldbach's conjecture would simply be that any even number is the sum of two primes. However, if we let 1 be a prime, then it would mean that the prime factorization theorem would not be true without adding a lot more wrinkles of its own. The prime factorization theorem says that any number can be uniquely factored into a product of prime numbers. So 6 = 2 x 3, and 20 = 2 x 2 x 5. We are so used to this that it doesn't seem amazing that it is true, but there is a sense in which it is surprising that, say, 51 = 3 x 17 and it can't also be 7 x something or 13 x something. The prime factorization theorem is so important that it has a second name, The Fundamental Theorem of Arithmetic. Here is a more rigorous definition: The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes If 1 was prime, we wouldn't need the little exception in parentheses. But then we wouldn't need the theorem at all, since it wouldn't be true. 6 would still be 2 x 3 but also 2 x 3 x 1 and 2 x 3 x 1 x 1 and 2 x 3 x 1 x 1 x 1 and so on. So going back all the way to Euclid, and probably before, 1 has been explicitly excluded from being a prime number. What Euclid proved was that if p is prime, and p divides A x B without a remainder, then either p divides A without a remainder, or p divides B without a remainder. The Fundamental Theorem of Arithmetic is a corollary of this, since it wouldn't be true if there was a number with two different factorizations. Fundamental Theorem of Algebra There is also a Fundamental Theorem of Algebra. However, it is not exactly the "Algebra I" you learned in middle school: The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root This is more important than it sounds. When you were in primary school, you learned about positive numbers: 1, 2, 3... Then you discovered that you can't solve 3 - 5 with only positive numbers, so you learned about negative numbers: -1, -354... Then you discovered that you can't solve 3 / 7 and so you learned about fractions, what mathematicians call rational numbers: 1/2, 3/4, 22/7... Then you discovered you can't solve x 2 = 2 with only rational numbers, so you learned about real numbers. 1.414... and 3.14159... Then you discovered, if you went far enough in math, that you can't solve x 2 = -1 so you learned about complex numbers: i , 1+2 i , and so on Then...nothing. The fundamental theorem of algebra tells you that you are done. You don't need to learn about some new type of number. Or, as mathematicians put it, the complex numbers are algebraically closed. I looked to see if there are any more interesting Fundamental Theorems I've never heard of, but Google only has these two. However, there is another really important fundamental concept... The Central Dogma of Biology As stated by Francis Crick back in 1958: The Central Dogma. This states that once 'information' has passed into protein it cannot get out again. In more detail, the transfer of information from nucleic acid to nucleic acid, or from nucleic acid to protein may be possible, but transfer from protein to protein, or from protein to nucleic acid is impossible. Information means here the precise determination of sequence, either of bases in the nucleic acid or of amino acid residues in the protein. This refutes Lamarckism, that an organism can pass on traits it learned to its offspring. It is more simply (but not 100% correctly) expressed as "DNA makes RNA, and RNA makes protein." Do You Know What Tau Is? I started this post saying that mathematicians had a choice as to whether they made 1 prime or not. Another place where mathematicians had a choice is the value of π (pi). It turns out that a lot of the time that π occurs in math, it shows up as 2π, starting with when you learned about the circumference of a circle as being 2πR. There is a whole subspecies of mathematicians who feel very strongly about this, and even have a letter τ (tau) for 2π. The math doesn't change just by renaming the concept, but a lot of math is simpler to express this way, from Fourier transforms, to the normal distribution (often called a bell-curve), and more. It even has a day, Tau Day, June 28th. If you want to go deep down the rathole, then read The Tau Manifesto , subtitle "No really, Pi is wrong.." Do You Know What Your Erdős Number Is? In my recent post on Zombies , I published an XKCD comic that doesn't make a lot of sense unless you know the story of Paul Erdős. He was the most prolific mathematician ever, publishing over 1,500 papers. He also had a notoriously weird lifestyle. He didn't live anywhere, he would stay at other mathematician's houses and say "my mind is open." Often, his host would explain some problem he'd been working on for a long time and Erdős would solve it almost instantly. They would then write a joint paper. The badge of honor for many mathematicians is to have a low Erdős number. Here's how it works. Paul Erdős has an Erdős number of zero. If you published a paper with him, you have an Erdős number of 1. If you published a paper with someone who has an Erdős number of 1 then your Erdős number is 2. And so on. So it's a sort of mathematical version of a Bacon number, as in the 6 degrees of Kevin Bacon game. There is even an Erdős-Bacon number, which is the sum of the two. One notable holder of a low Erdős-Bacon number is actress (and mathematician) Danica McKellar, who has an Erdős number of 4 and a Bacon number of 2, making a total of 6. President's Day Puzzle This isn't anything to do with Presidents, it's not one of those puzzles that looks like it's mathematical but is not: 16 20 25 35 ? (these are the numbers of the US Presidents who were assassinated—hopefully, the "?" never gets filled in). So here's your Presidents' Day puzzle. What comes next? 110 20 12 11 10 ? Sign up for Sunday Brunch, the weekly Breakfast Bytes email.![]()
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