see things differently

Posts tagged with √2

1729 is not a number

SOURCE: Barry Mazur, When Is One Thing Equal to Some Other Thing?

Anyone who writes a number in the form 1729 implies a method of calculation: one thousand, plus seven hundreds, plus two tens, plus nine ones.

$& \underline{\texttt{1729}} \\ & \ 1 \cdot 1000 \\ + & \ 7 \cdot 100 \\ + & \ 2 \cdot 10 \\ + & \ 9 \cdot 1 \\$

Different than writing tick-marks |||||||||||||||||||||||…¹⁷²⁹ which would imply

$\underbrace{1 + 1 + 1 + 1 + 1 + 1 + 1+ \ldots}_{\text{1729 times}}$

Different than Roman numerals MDCCXXIX,

$& \ \; \underline{\texttt{M\!\;\!D\!\;\!C\!\;\!C\!\;\!X\!\;\!X\!I\!X}} \\ & \ \; 1 \cdot 1000 \\ + & \ \; 1 \cdot 500 \\ + & \ \;2 \cdot 100 \\ + & \ \; 2 \cdot 10 \\ - & \ \; 1 \cdot 1 \\ + & \ \; 1 \cdot 10$

hexadecimal 6C1,

$& \ \; \underline{\texttt{6C1}} \\ & \ \; 6 \cdot 256 \\ + & \ \; 12 \cdot 16 \\ + & \ \; 1 \cdot 1 \\$

or the most agnostic way to write a number, via its prime factorization ”the fourth prime ⨯ the sixth prime ⨯ the eighth prime”.

$p_4 \cdot p_6 \cdot p_8 \text{ \ \ i.e., \ } 7 \cdot 13 \cdot 19$

They’re all ways of calculating the number, but they’re not the number itself.

Daphne

We could agree to call this number some agreed-upon name, like ”Daphne” and use a symbol ₯ for shorthand. 1729 is no more her name than is 6C1.

Or we could refer to Daphne by property without implying a particular calculation: “the smallest sum of two cubes, which can be written two different ways”.

Or we could denote Daphne by equation:

₯ = 9^3 + 10^3, or
Daphne ₯ is the number that solves the equation 12^3 - ₯ = 1^3.

It’s the same way with the square root of two. Its name is no more √2 than ⨿2 or ¶2.

Just like 1729, √2 is merely a notation. What makes √2 be √2 is the property it has.

**Numbers aren’t numerals, they’re … uh … things.**

All of the above is meant to drive a wedge between numbers as written on paper and numbers as they “exist” abstractly.

Numbers don’t need numerals. And you can talk about numbers without knowing how to write them. Just agree on some symbol like π and use π whenever you want to talk about the number you don’t know how to write.

It sounds trivial talking about an integer, but the difference between

properties of numbers,
ways to calculate numbers, and
the numbers themselves

is good to keep in mind when you’re thinking deeply about

the real line (measure theory),
algebraic numbers (Galois theory),
transcendental numbers,
p-adic numbers,
complex numbers or other bodies of numbers.

sqrt10

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Pythagorean Theorem

This is how I first really understood the Pythagorean Theorem.

The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.

Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.

Ideas involved here:

scaling properties of squared quantities
(gravitational force, skin, paint, loudness, brightness)
circumcircle & incircle
√2

This is also how I first really understood √2, now my favourite number.

hi-res

Pythagoras

visual proofs of the Pythagorean Theorem, a² + b² = c²

proof 3

proof 2

proof 4

hi-res

Here’s the distribution of the first million digits of the square root of two’s decimal expansion.

Number of digits | is: 0's | 99 818 1's | 98 926 2's | 100 442 3's | 100 191 4's | 100 031 5's | 100 059 6's | 99 885 7's | 100 012 8's | 100 347 9's | 100 126

If each digit had a Bernoulli chance of coming up (like a 10-sided die), you’d expect to see 10 000 ± 30 times. And going on with that same assumption, the chance of the least-frequent digit coming up less than 99 000 times would be something like one percent.

What does it mean? I will meditate on this and expand √2 in different bases besides 10.

hi-res