What is the smallest set of letters that can spell any integer?

Question

Today my 9-year-old nephew told me that he can spell any integer in English using only 9 letters. This is how he's doing it:

    ...
-3: MINUS ONE MINUS ONE MINUS ONE
-2: MINUS ONE MINUS ONE
-1: MINUS ONE
 0: ONE MINUS ONE
 1: ONE
 2: ONE PLUS ONE
 3: ONE PLUS ONE PLUS ONE
    ...

The letters he's using are E, I, L, M, N, O, P, S and U.

Can we do better and spell any integer with less than nine letters? I think so!

Just please be sure to briefly explain any maths you are using in your method, so that my young nephew (and his uncle) can understand.

Answer 1

  ...
-3: MINUS ONE MINUS ONE MINUS ONE
-2: MINUS ONE MINUS ONE
-1: MINUS ONE
 0: ONE MINUS ONE
 1: ONE 
 2: ONE MINUS MINUS ONE
 3: ONE MINUS MINUS ONE MINUS MINUS ONE
    ...

two letters fewer since we do not use PLUS any more.

Answer 2

4 letters: E N O T

Explanation:

My idea is to use binary and marcoresk's idea of using NOT to represent the - sign, although I have to admit it might be slightly 'cheating' as I have to define things and work in a different base.

To represent a number you read out its binary representation using ONE and NONE to represent $1$s and $0$s.

Eg. $5$ would be: ONE NONE ONE

If the number is negative, add a NOT to the beginning.

Eg. $-6$ would be: NOT ONE ONE NONE

Extension:

You can represent any decimal which is terminating in binary with an extra letter D for a total of 5 letters which allows you to say DOT for the decimal point.

Eg. $-5.5$ would be: NOT ONE NONE ONE DOT ONE

Other possibilities:

I understand that using NOT probably isn't the best idea to indicate -, so as I've mentioned in the comments, perhaps NEG can be used instead as it's an abbreviation for negative (I wasn't sure of this until Jason C pointed it out in his answer). This will still result in 4 letters being used (although 6 for the extension).

Answer 3

If we use LESS to indicate subtraction, then we have

-1 = ONE LESS ONE LESS ONE

0 = ONE LESS ONE

1 = ONE

2 = ONE PLUS ONE

etc., using only seven letters ELNOPSU.

Answer 4

If any integer can be spelled with this structure

one some operation one some operation one

the trick could be simply this: reduce the numbers of letters needed for operations (or use O, N, E as much as you can!).

M,I,N,U,S introduces 4 new letters (N does not count) P,L,U,S introduces 2 other letters (P and L)

A simple (but symbolic) solution is to represent a operation with a single letter... but this is not a way to "say it in English".

So I propose this structure: PLUS becomes AND (only two more letters, A and D). Every integer number N bigger than 1 can be said as

N =  ONE AND ONE AND ONE AND ONE...

B) Define the number MINUS ONE as "NOT ONE" (let's say this will became our convention, this is the weakest part!) in order to introduce only one more letter (T)

Now zero becomes

ZERO =  ONE AND (NOT ONE)

C) Since every negative numbers is equal to its absolute value multiplied by -1 e.g. -59 = 59 * (-1) I suggest to use a new word to introduce the operation of multiplication, as in class we often say A multiplied by B as "A dot B" (A dot is one possible symbol to show multiplication in advanced math, instead of x or * or others)

Note that now DOT does not introduce any new letter. So -3 could be

(ONE AND ONE AND ONE) DOT (NOT ONE)

Notice we have used only 6 letters instead of 9 (and mantaining a speakable structure) O, N , E, D, T, A

Now the last problem: how to "say" parenthesis? With the rithm of speech! ONE AND ONE AND ONE AND ONE [after this sequence do a little pause] DOT [say DOT as an important word, then a little pause] NOT ONE [say NOT ONE fast, as if would be a single word]

Thank you for this pretty puzzle and sorry for my poor english!

Answer 5

Since

"O" is in the Collins English dictionary as "used to mean zero"
(pronounced the same as "owe", as in "I owe MackTuesday a coconut")

...and, as pointed out by @MackTuesday, "a" means one
(as in "I owe MackTuesday a coconut")

We can

Use just two letters: O, and a
(I previously used three with One rather than a, which may be less confusing)

If we use base 2, or binary with a signed magnitude representation by using our first bit as an indication of "is this number negative" and the time we stop speaking to represent our most significant bit for example:
$-32 =$ a O O O O O a $=- (0\times 2^0 + 0\times 2^1 + 0\times 2^2 + 0\times 2^3 + 0\times 2^4 + 1\times 2^5)$

$-7 =$ a a a a
$=- (1\times 2^0 + 1\times 2^1 + 1\times 2^2)$

$-4 =$ a O O a
$=- (0\times 2^0 + 0\times 2^1 + 1\times 2^2)$

$0 =$ O O (or a O)
$=+ (0\times 2^0)$ (or $=- (0\times 2^0)$)

$17 =$ O a O O O a
$=+ (0\times 2^0 + 0\times 2^1 + 0\times 2^2 + 0\times 2^3 + 1\times 2^4)$

Note that

$2^n$ just means $2$ multiplied by itself $n$ times, so $2^3$ is $2\times 2 \times 2=8$
...and that the identity element of the multiplication group is $1$ so $2^0$ is $2$ multiplied by itself $0$ times, which must be $1$.

Answer 6

6 Letters: T I M E S U

-1: i times i
0: sum i i times i times i
1: i times i times i times i
2: sum i times i times i times i i times i times i times i
3: sum i times i times i times i i times i times i times i i times i times i times i
-2: i times i times sum i times i times i times i i times i times i times i

i being the mathematical constant for the square root of -1. i squared ("i times i") is -1 and i to the fourth power ("i times i times i times i") is 1.

If the pause between the items in the set it not clear enough, you could also use the notation:
3: sum set item i times i times i times i item i times i times i times i item i times i times i times i

Answer 7

Another way of using only seven letters (A, D, E, L, N, O, S):

    ...
-3: ONE LESS ONE LESS ONE LESS ONE LESS ONE
-2: ONE LESS ONE LESS ONE LESS ONE
-1: ONE LESS ONE LESS ONE
 0: ONE LESS ONE
 1: ONE
 2: ONE AND ONE
 3: ONE AND ONE AND ONE
 4: ONE AND ONE AND ONE AND ONE
    ...

Answer 8

ONE ON ONE = 2

ONE OFF ONE = 0

ONEF: total 4 letters

by replacing plus and minus with on and off which mean the same.

Answer 9

If we are allowed any encoding we like, we only need one letter. We choose some way of enumerating integers (say, in the order 0, 1, -1, 2, -2, 3, -3 ...) and then we encode the $n^\mathrm{th}$ integer as $n$ repetitions of our letter.

I don't think this is in the spirit of the question, which means that many of the other answers also aren't in the spirit of the question ("spell any integer in English"; I interpret this to mean, if you need to explain what number you are spelling, you have already lost). But if you're going to give an unspirited answer, you might as well give the optimal one :)

Answer 10

The rules to this seem a little ambiguous but does this count? 4 letters:

{O,N,E,G}

Where all positive integers are just a that many ones:
1: one
2: one one
3: one one one


And negative integers are indicated by "neg" (an English abbreviation):
-1: neg one
-2: neg one one
-3: neg one one one


And zero is just an o:
0: o

Answer 11

5 (O N E L S)

For example: 110=ONE ONE ONE LESS ONE=111-1 Just do the lowest integer above your target value consisting of only ones, then subtract down. Works for any positive or negative number.

Answer 12

2 answer depends on the rule

write numbers in binary representation (9 letters o,n,e,z,r,m,i,u,s) 1 = one
-1 = minus one
0 = zero
2 = one zero

or

or maybe just 6 letters (m,i,n,u,s,o) if we can write 1 as i and 0 as o (maybe invalid) 1 = i
-1 = minus i
0 = o
2 = io

Answer 13

We can refer to the number 1 using the less common word UNIT. Using MINUS adds two letters. Then using the word SUM in place of PLUS adds no new letters. Unlike PLUS, which is an operation between two operands, SUM naturally operates on all following operands.

-3: MINUS SUM UNIT UNIT UNIT
-2: MINUS SUM UNIT UNIT
-1: MINUS UNIT
 0: UNIT MINUS UNIT
 1: UNIT
 2: SUM UNIT UNIT
 3: SUM UNIT UNIT UNIT

If you're unhappy with the SUM operator taking an arbitrary number of operands, we can have it apply to only two operands. I've summarised below, using Polish notation for both SUM x y=x+y and MINUS x y=x-y.

-3: MINUS UNIT SUM SUM SUM UNIT UNIT UNIT UNIT
-2: MINUS UNIT SUM SUM UNIT UNIT UNIT
-1: MINUS UNIT SUM UNIT UNIT
 0: MINUS UNIT UNIT
 1: UNIT
 2: SUM UNIT UNIT
 3: SUM SUM UNIT UNIT UNIT

Thus we have 6 unique letters: IMNSTU

Answer 14

Five letters: {Z, E, R, O, N}

No maths required, but you have to understand binary. We spell out the bits involved in a binary representation of the desired integer. The first bit is the sign bit (ZERO for positive, ONE for negative).

Examples:

-3: ONE ONE ONE
-2: ONE ONE ZERO
-1: ONE ONE
 0: ZERO ZERO
 1: ZERO ONE 
 2: ZERO ONE ZERO
 3: ZERO ONE ONE

Answer 15

I used only SUB instead of PLUS and MINUS.

Using only six letters (B, E, N, O, U, S):

-3 : SUB (ONE ONE ONE)

-2 : SUB (ONE ONE)

-1 : SUB (ONE ONE)

0 : (ONE SUB ONE)

1 : SUB (SUB ONE)

2 : SUB (SUB ONE SUB ONE)

3 : SUB (SUB ONE SUB ONE)

Answer 16

Looking at all the number in binary give use the word zero and One which results in 5 characters.

8 = 1000 = one zero zero zero.

15 = 1111 = one one one one.