The Six Symbols

Question

The professor turned to me and said, "I believe that all numbers in mathematics can be expressed with only 6 symbols."

"A bold claim," I replied. "I suppose the you want all numbers to be encoded in base 6."

"Not at all. I am referring to Base 10."

"Then how is such a thing possible?" I questioned. "There are 10 digits, and even those aren't enough to let you express decimals!"

"On the contrary. See, the symbols I have chosen are not limited to numbers. They could include things like an addition sign, or possibly an exponentation sign."

"But even if you choose your symbols carefully, could you express every rational number? What about e? What about i or pi? What about sines, roots, and logarithms? Math is too complicated to be expressed in only six symbols!"

"You misjudge my work. Everything you have just mentioned can be expressed with my 6 symbols!"

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What were the six symbols the professor chose?

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Notes:

• We will assume Base 10 and standard notation for this question.

• The symbols are all fairly simple, and do not include things like summation or randomness.

• The following can be expressed with the symbols:

  • Any rational number

  • e, i, and pi.

• You can perform the following operations on the above numbers:

  • Addition

  • Subtraction

  • Negation

  • Multiplication

  • Division

  • Exponentation

  • Rooting

  • Sine, cosine, and tangent

  • Log with any base

• This means you can combine any of these. For example, you could express √(2*pi).

• The answer is very simple, although not very obvious. If your answer seems like a cheat, it's probably not correct.

• The final goal is to be able to express all combinations of the above numbers and operations. Infinity is not included, and infinitely long sequences are neither.

Answer 1

I believe we can do everything described in the question with the following symbols:

1, -, exp, log, and parentheses.

So, here's how it goes. First of all,

0 is 1-1; then x+y is x-(0-y); now xy and x/y are exp(log x +- log y). So we have all the usual arithmetic operations and can therefore form all the rational numbers.

Now

e is exp(1), i is exp(1/2 log(-1)), and i pi is log(-1).

From this we get

cos x in (exp(ix)+exp(-ix))/2, sin x = (exp(ix)-exp(-ix))/2i. Tan is the ratio of these two.

For exponentiation and rooting

we have x^y = exp(y log x) and y'th root of x = exp(1/y log x).

For log to arbitrary base

$\log_b(a)=\log(a)/\log(b)$.

I'm pretty sure this gives

the right branch cuts for exponentiation and rooting

but I'm a little uneasy about

multiplication and division in the presence of zeros: we need log to give "$-\infty" in that case, and exp to accept that as input and yield 0.

Answer 2

There is no solution if infinite strings are not allowed.

There are a countable number of finite strings on an alphabet of 6 symbols, and an uncountable number of complex numbers. There is no surjection from a countable set to an uncountable one.