Base $10$ uses these digits: $\{0,1,2,3,4,5,6,7,8,9\};\;$ base $2$ uses: $\{0,1\};\;$ but what would base $1$ be?
Let's say we define Base $1$ to use: $\{0\}$. Because $10_2$ is equal to $010_2$, would all numbers be equal?
The way I have thought Base 1 might be represented is tally marks, $0_{10}$ would be represented by nothing. So, $5$ in Base 1 would be represented by $00000$? Or we could define Base 1 to use: $\{$|$\}$ and $5$ would be |||||?
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Yes, the usual answer is that numbers are represented by "tally marks" in base $1$. However, the numeral $0$ might not be the best choice of a tally mark because if $00000$ were interpreted in base $1$ analogously to its interpretation in other bases, then it would be interpreted as $0 \cdot 1^5 + 0\cdot 1^4 + 0 \cdot 1^3 + 0 \cdot 1^2 + 0 \cdot 1^1$, which is $0$ rather than $5$.
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I would like to expand on Trevor Wilson's answer. Base-$b$ representation of integers is rooted in the fact that, for any non-negative integer $n$, there is a unique representation of $n$ in the form $$n = \sum_{i=0}^\infty a_ib^i$$ where $0 \le a_i < b$. For example, when $b$ is 3, and $n$ is 47, the unique solution has $a_0 = 2, a_1 = 0, a_2 = 2, a_3 = 1, $ and $a_i = 0$ for all $i>3$. The $a_i$ are called the "base-$b$ digits of $n$"; in our example the base-3 digits of 47 are 1202. We say that the sequence of digits is a numeral, and that it represents the number $n$.
The uniqueness property means that each $n$ has exactly one base-$b$ representation. If one requires that the sequence of $a_i$ is eventually zero (that is, that $a_i = 0$ for all sufficiently large $i$) then the converse holds also: each sequence of digits corresponds to exactly one $n$. In fact there are four properties that hold:
It is quite possible to construct representations that lack some of these properties. For example, consider the base-3 representation, but drop the restriction that says that $0\le a_i < 3$. Then property 2 fails: The number 47 has many base-3 representations: 502, for example, or 362, or 1 12 2 (here $a_1 = 12$), or even one (harder to write) where $a_0 = 47$. Each sequence of digits still represents a single $n$, but a particular $n$ might have many representations as a sequence of digits. Sometimes such representations even have some use.
Some of these properties are more important than others. Property 4, for example, is crucial, because if it doesn't hold, then there is some sequence of digits that might represent two different numbers, and when you see it you don't know what number is being represented. Such a system can't really be called a system for representing numbers.
Similarly, a system which fails to have property 1 has limited usefulness. Such a system can represent some $n$, but not all.
Depending on where and how it fails, a representation might be more or less useful. Fraction notation, for example, is universally used to represent rational numbers. But it fails to have properties 2 and 3! (It fails 2 since each rational number has many representations, say as $\frac12, \frac24, $ or $\frac{120}{240}$. And it fails 3 since $\frac10$ and $\frac00$ do not represent any rational numbers.) But these failures don't prevent it from being useful as a representation of rational numbers. A more serious failure arises if you try to make fractions represent real numbers; then property 1 fails, since there is no fraction representation for the number $\pi$ or $\sqrt2$.
Now let's return to $$n = \sum_{i=0}^\infty a_ib^i.$$ I said that this representation of non-negative integers has all four properties, but I left out an important limitation: the four properties only hold for $b\ge 2$. If $b=1$, the restriction $0\le a_i<b$ degenerates to $a_i=0$, and we can no longer represent any number except 0. So only 0 has a base-1 represenation. As a numeral system, this is completely useless.
If we drop the $0\le a_i<b$ restriction, we get something that hardly resembles a system of representation at all: Each number $n$ now has many base-1 representations For example, one could write 5 as 14, or 32, or 1121.
So, although it is inconsistent, mathematicians, and especially computer scientists, adopt a different meaning for "base-$1$ representation". They abandon $\sum a_ib^i$ completely and agree to represent the number $n$ as a sequence of exactly $n$ ones. For example, $7$ is represented as 1111111. This restores properties 1–4, so it is a sensible representation.
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There is no base $1$, and no unary number system. Base $b$ requires at least two symbols from $0$ to $b - 1$. Base $b$ does not use the digit $b$. For instance base $2$ does not use the digit $2$. So any system that uses the digit $1$ cannot be base $1$.
Tally marks are typographic representation of integers, but are not a "base", let alone "base 1".
The printed representation of a number in a base has a length which is proportional to the logarithm of the number. In the tally mark system, the length is proportional to the number.
Bases can represent fractions. For instance 1.11 in binary is one and three quarters. This works thanks to the negative powers to the right of the point. If we multiply one and three quarters by two, we can simply move the binary point: 11.1. By golly, this is three and a half, exactly right.
The best that the tally system can do here is to be repaired with some scheme whereby tally marks after a dot represent an enumeration of the countable set of fractions. For instance .1 could mean half, .11 means a third, .111 two thirds, and so on. But this scheme is fundamentally incompatible with what is on the left of the point. Moving the point has no intuitive meaning.
The use of at least two symbols in bases is related to the field of numbers having two elements: an additive identity (zero) and multiplicative identity (one). The naive tally mark system ignores this concept of a field with two identity elements, which is why it runs aground when it comes to representing fractions. It doesn't even represent zero, except perhaps by means of leaving an ambiguous empty space devoid of tallies to represent nothingness.
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Though many consider a tally system to be a base 1 system, I think this confuses more than it clarifies. A tally system is not a radix system and does not represent numbers using the same mechanism. The tally system is simply a string whose length represents the number (integer) in question while a radix numeral is a shorthand for a power series.
Therefore to imply that each of these distinct systems have a comparable attribute known as their "base" is misleading. Much like claiming that a playing card deck and a computer plotter each has a comparable operation called "draw".
You're exactly right that such a system would be represented by the use of arbitrary tally marks. Such a system is known as a Unary Numeral System (Wikipedia Entry):
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