When you were first taught how to write numbers and them together, you were probably told something about the “ones place”, the “tens place”, the “hundreds place”, and so on. Each further to the left represents a larger of ten, and the multiples are added together to get the number you want—so \(8207\) is counted as \((7 \times 10^0) + (0 \times 10^1) + (2 \times 10^2) + (8 \times 10^3)\), or “seven ones, zero tens, two hundreds, and eight thousands”. But why use powers of ten? What’s so special about that number?
Well, nothing, actually. We probably only use powers of ten because we happen to have evolved with ten fingers—any number bigger than one will work just as well, although it may look strange at first. Since two is the smallest integer larger than one, a binary number notation—a notation that uses powers of two—is one of the simplest possible. noteThe word “binary” has the same root as the word “bicycle”—bi meaning “two”.
Binary notation uses only two digits, \(0\) and \(1\), and each “place” to the left goes up by a power of two instead of ten (in other words, it doubles). For example, in binary notation, the number \(11010\) is counted as \((0 \times 2^0) + (1 \times 2^1) + (0 \times 2^2) + (1 \times 2^3) + (1 \times 2^4)\), or “zero ones, one two, zero fours, one eight, and one sixteen”. Translating back to the familiar ten, we would write it as \(26\).
You may notice that binary notation tends to be a bit longer than decimal (for example, “11010” takes more characters to write than “26″). It’s also more difficult to read, unless you have a lot of practice with it. So why would anyone use it? Well, for one thing, binary notation is often very convenient for talking about powers of two, for instance when working with base-two logarithms or log odds, or when working with some quantity measured in bits. It’s also essential for working with computers, as all modern computers store and manipulate data, on the lowest level, exclusively using binary notation.
Mathematics is the study of numbers and other ideal objects that can be described by axioms.