A binary number denotes a sum of powers of two. 10102 means (from right to left) no one, a two, no four, an eight.
I'll try to give a more expansive explanation of the concept of bases here.
I would argue that a number is an abstract concept. When you read
10, it is not the real thing, it is merely one possible representation of an idea/concept that you cannot truly grasp.
10 (that is,
1 followed by a
0), is an encoding for this idea of "ten". Although we have may have developed some sort of intuition for this way of encoding numbers, there is a more fundamental system at work here.
143 for instance is
100 + 40 + 3, or: 3 ones, 4 tens and 1 hundred (ten tens). We can decompose any such number by their digits. The position of a digit determines its significance, i.e. what power of ten it counts:
As you can see, the digits denote the coefficients for a sum of powers of ten. We call this number (ten) the base.
Ten (as a concept) is not a special number. It just seems natural because we are so used to it.
We can freely choose virtually any basis we like. The meaning of each digit always depends on the base and is obtained as follows:
143 (the encoding) can have a different meaning, depending on which base is implied (usually ten of course) but the base could just as well be 11:
With binary numbers, it works the exact same way, but the base is two of course.
For any encoding in a certain base, you need base-many distinct symbols to encode numbers. That is, for base ten, you need ten distinct symbols, for a base-two system, you need two distinct symbols, or for hexadecimal, you need sixteen distinct symbols. Also, you need to have the idea of a "successor", i.e. in the decimal system, the symbol "2" is the successor of the symbol "1".
In a system that uses the the special symbols
1 (with their common meaning),
10 is always the encoding for its base.
"common meaning": 0*a = 0, 1*a = a
This system can be used to denote any objects that are enumerable.
In real world terms - visually
Try to imagine a series of objects starting at some point, continuing indefinitely.
For the purposes of this document, I'll use dots:
Zero . . . . . . . . . …
Every dot represents a number. naturally, each dot is the successor of the dot to its left.
One way to refer to a dot would be to assign a unique symbol to each dot. As you probably can imagine, without any system, this is going to be extremely cumbersome. Just to be able to count to one thousand, you would have to remember one thousand arbitrary symbols (the Chinese for example prove it's possible, but that's beside the point).
Let's just go with the assumption that it's cumbersome, and provide some shortcut: I'll just replace every fourth dot with a pipe:
Zero . . . | . . . | . . . | . …
Now, we have simplified the problem a bit, assuming we can reliably name which pipe we mean, we can name dots relatively to a pipe. For instance, I can say: The second point after pipe X. Or the second point after the third pipe to denote the number fourteen.
However, as numbers get bigger (or speaking visually: we zoom out, we are left with a series of pipes), it's extremely cumbersome to simply count the pipes and then refer to a dot relatively of one such pipe. We apply the same trick. We can replace every fourth pipe with an ampersand:
Zero . . . | . . . | . . . | . . . & . . . | . . . | . . . | . . . & . . .…
Now, it becomes easier to specify a certain pipe, we simply say which ampersand we mean. Going from that, we relatively specify the pipe, and going from that, we relatively specify the dot.
This is exactly what the base-notation does. It places these markers on every n-th object, different markers on every n-th marker, etc Thus, there are never n or more of the same markers in sequence.
In this case, we have encode numbers with the base four: 1234 refers to the first ampersand, the second pipe after that, and the third dot after that: The number twenty-seven.
Applying the aforementioned rule, we never need more than four distinct symbols to denote an arbitrary number in base four.
Proof: Let's assume we need more than four distinct symbols. This would imply, that I need to count four or more markers of the same kind from a given position. However, that is never necessary, since I replaced every fourth of these very markers with a different kind, I must be able to skip ahead. Thus, the premise is incorrect: Four distinct symbols are always enough.