Best way to express 2014

Question

In my class, I always write the date using mathematical formulas, or cool little equations. I want to show my students that even the most mundane seeming number often has fascinating features, and its own beauty - that's the reason I got into mathematics, and I want to pass it down.

For example, for 30, I wrote $\frac{6!}{4!}$.

Now, 2014 has kind of stumped me. The best I could come up with was this, using factorials:

$$(2!(2!+2!(4!))+3!)(4!-3!+1!)$$

But, this isn't the most attractive, or interesting, equation.

What do you think the best (nerdiest?) way to write 2014 is?

"Best" - shortest, more concise. "Nerdiest" - using mathematical functions in interesting ways. — James Williams, Mar 07 '14 at 18:18
The shortest and most concise way to write $2014$, I suspect, is to write $2014$. A nerdy twist? express in terms of its prime factors: $2\cdot 19\cdot 53$ — amWhy, Mar 07 '14 at 18:19
Ahah yes, probably should have put "shorter" there... You know what I mean! — James Williams, Mar 07 '14 at 18:21
You could play around with different bases: it's $a3c_{14}$, or $7de_{16}$. Another nice one is $bbc_{13}$. — user88595, Mar 07 '14 at 18:22
@user88595 Just had a quick look at that, two good answers on there, thankyou — James Williams, Mar 07 '14 at 18:44
@JackM: Definitely make that an answer -- it's IMO the best one so far. — Ilmari Karonen, Mar 07 '14 at 19:30
Did you know that 2014 is the only natural number that precedes 2015 and succeeds 2013? — evil999man, Mar 08 '14 at 18:04
In Austrian german, we have a word for it. This year = heuer. — Nikolaj-K, Mar 08 '14 at 22:56
Not sure why this question has been put on hold. Yes, it is primarily opinion based, but it has been tagged as a soft question - there is no need for solid answers here, I'm just having a bit of fun. Granted, this question may not fit the "site format", but really, who cares? The question has generated lots of interesting, math based, responses - how is this not a fit for the site? — James Williams, Mar 10 '14 at 06:23

score 101 · Answer 1 · answered Mar 07 '14 at 19:45

101

Looking $2014$ up in the OEIS turns up:

$$2014=13^3-13^2-13^1-13^0$$

In general, looking a number up in the OEIS is probably a reasonable way to turn up pleasing identities.

answered Mar 07 '14 at 19:45

Jack M

27,819
7
63
129

2

Use the keyword:easy to get simple results like this. – Ypnypn Mar 09 '14 at 19:25

score 42 · Answer 2 · answered Mar 07 '14 at 19:39

42

Give them the following :

$$(-2+0+1+4)^{(2+0+1+4)}-(2+0+1+4)^{(-2+0+1+4)}+(2+0-1+4)^{(-2+0+1+4)}+(2+0-1+4)\cdot(-2+0+1+4)^2=?$$

and tell them to compute the result.

All you can see is only 2014 with some sign changed and of course the result is simply
$$3^7-7^3+5^3+5\cdot3^2=2014$$
It will look better on a board.

answered Mar 07 '14 at 19:39

Konstantinos Gaitanas

9,115

1

I like this one; With all due respect, but many of the other answers don't look more interesting to me than 2⋅10³+0⋅10²+1⋅10¹+4⋅10⁰. – Mr Lister Mar 08 '14 at 10:51
@MrLister thank you very much for the comment – Konstantinos Gaitanas Mar 08 '14 at 13:46

score 34 · Answer 3 · edited Jun 12 '20 at 10:38

34

With all due credit to this base 13 answer on codegolf.SE:

$$2014=BBC_{13}$$

Or just playing with my calculator, I like the look of

$$2014=5^5-1111$$

edited Jun 12 '20 at 10:38

Community

1

answered Mar 07 '14 at 19:30

Digital Trauma

193

score 21 · Answer 4 · answered Mar 07 '14 at 18:31

I hope you will enjoy the following spoof :

$\qquad\quad$ I remember once going to see him for the Holidays, and remarked that the number of the upcoming year seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number which can be expressed as the product of three distinct primes, which are congruent modulo $17$." $:)$

score 18 · Answer 5 · answered Mar 07 '14 at 18:23

What it lacks in brevity, it makes up for in nerdiness:

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS$0$

I like it ;) although I have a feeling that writing this out on a board could be rather monotonous — James Williams, Mar 07 '14 at 18:26
@K.Rmth well, it appears to be 2014 S's and 1 zero. Not sure what the 0 is for (or why S was chosen) — Tyler, Mar 07 '14 at 19:59
@K.Rmth $S$ is the successor function. So, for example, $S(4) = 5$. In general, $S(x) = x + 1$. — Gamma Function, Mar 07 '14 at 20:00
Peano arithmetic has one constant symbol, 0, and one unary function symbol, $S$. Every natural number is either $0$ (in which case it is not the successor of anything) or the successor of some (unique) other natural number. Every natural number can thus be written as a finite number of applications of the successor function to the constant zero. Thus the natural numbers are defined recursively, which makes this essentially the minimal structure needed to enable inductive type proofs. (I'm simplifying slightly - some formulations of Peano Arithmetic permit nonstandard natural numbers.) — Unwisdom, Mar 07 '14 at 20:23

Yiyuan Lee · Answer 6 · 2014-03-07T18:30:15.247

13

Here are some cryptic ones: From the Gaussian integral we have $$2014 = \frac{4028}{\sqrt{\pi}}\int_{0}^{\infty}e^{-x^2}dx$$ and from the Basel Problem we have: $$2014 = \frac{12084}{\pi^2}\sum_{i=1}^{\infty}{\frac{1}{i^2}}$$ Here are some that (arguably) has deep meanings and roots: $$ 2014 = 2\cdot19\cdot53$$ $$2014 = 2^{11} - 34$$ For some trigonometry we have: $$2014 = \frac{4}{\cos^3{\frac{\pi}{9}}\cdot\cos^3{\frac{2\pi}{9}}\cdot\cos^3{\frac{4\pi}{9}}} - 34$$ It depends on perception, really. There are probably arguably infinitely many ways to write $2014$ in a "short, snappy, cool, and nerdy way".

edited Mar 07 '14 at 18:30

answered Mar 07 '14 at 18:16

Yiyuan Lee

14,435

You definitely have the right idea here! I am intrigued as to how you worked out the first two in such a short time – James Williams Mar 07 '14 at 18:23
1

@JamesWilliams Well, you have 22014=4028 and 62014=12084, where the 2 and the 6 belonged to the (rather well known) cases where the 2014 is removed from both sides, and rearranged so the sum/integral are on their own side. – FireGarden Mar 07 '14 at 19:00
2

This answer misses the point. These identities don't reveal anything interesting about the number 2014. They would work for any number. – user85798 Mar 10 '14 at 05:29

Barry Cipra · Answer 7 · 2014-03-07T18:43:29.020

13

How about

$$3\cdot6!-5!-4!-2!$$

or, if you like

$$(6!-5!)+(6!-4!)+(6!-2!)$$

Alternatively:

$$6!2!+4!4!-2!0!$$

edited Mar 07 '14 at 18:43

answered Mar 07 '14 at 18:28

Barry Cipra

79,832

score 11 · Answer 8 · answered Mar 07 '14 at 18:53

11

(Self answering question) Find the integral part of the unique real root of the equation $$\log_2 x+\log_{20}x+\log_{201}x+\log_{2014}x -2-0-14+\frac{1}{20+\frac{1}{14}} = 0$$

answered Mar 07 '14 at 18:53

Lord Soth

7,750
20
37

6

Using "2014" in the expression itself is surely cheating. – David Richerby Mar 07 '14 at 22:45

diesirae · Answer 9 · 2014-03-08T12:39:55.713

11

$$ 2014 = 2^{2\times2^2 \times (2\times2^2-2)}-2\times (2\times2)^2-2 $$

Equations like this can be made for any number, not just 2014.

edited Mar 08 '14 at 12:39

answered Mar 08 '14 at 12:34

diesirae

1

score 6 · Answer 10 · edited Mar 25 '15 at 21:11

6

Using a base 2014 number system, it would be expressed as:

10

edited Mar 25 '15 at 21:11

ahorn

2,236

answered Mar 09 '14 at 23:24

FerretallicA

101

score 5 · Answer 11 · answered Mar 07 '14 at 18:46

5

Maybe do:

$$2014 = \sum_{k=0}^{11}\binom{11}{k}-34$$

answered Mar 07 '14 at 18:46

Patrick

2,106

1

aka 2^11 - 34, since 2^11 = 2048 – smci Mar 08 '14 at 18:06
I like my way better :-) – Patrick Mar 08 '14 at 21:44
1

I was just pointing out why they're equivalent... – smci Mar 09 '14 at 15:04

burfor7 · Answer 12 · 2014-03-08T01:46:35.593

Binary: 11111011110 Hexadecimal: 7DE

Image the students perplexing expression when they see: 1101/11/111111011110 or C/3/7DE Tell them to write this date in this form on there notes. Guaranteed they will show it to their friends or family.

Wow I can't believe I haven't thought of doing this with my students. As a rookie high school math teacher I am always looking for new 'hooks' with my students. Great idea and thank you!

Lesson planning using stack exchange? Who knew..

score 2 · Answer 13 · edited May 23 '17 at 12:39

2

How about multiple mathematical formulae (using only digits and simple operators) for every year (except 2102) for the rest of the century? e.g.,

$$2014 = 10*9*8*7/6/5*4*3-2*1$$

Read more about this in the fascinating Wolfram Blog. To generate these equations, take a look at the answers for this SO question.

edited May 23 '17 at 12:39

Community

1

answered Mar 09 '14 at 18:07

lifebalance

111
5

score 2 · Answer 14 · answered Mar 10 '14 at 02:27

2

I would express this as a subtraction of powers of two, i.e. $$ 2^{11} - 2^5 - 2^1 $$

answered Mar 10 '14 at 02:27

Mauren

101

score 1 · Answer 15 · answered Mar 09 '14 at 23:30

If anyone went to HCSSiM, we have a tradition of worshiping the number $17$.

So I would say the coolest way to express $2014$ is $$2014=2 \times 19 \times 51$$

It is easy to show that this neat fact: $2014$ is in fact the smallest number that can be expressed as a product of three distinct positive integers that are all congruent modulo $17$.

Best way to express 2014

15 Answers15