The Binary Number System
Unlike the decimal system, which uses ten digits, the binary system uses just two symbols:
\[\mathsf{0 }\textsf{ and } \mathsf{1}\]
Technically, any two symbols could have been chosen. But since we were already familiar with the decimal system, we adopted the first two digits, \(\mathsf{0}\) and \(\mathsf{1}\), for convenience.
The Influence of Leibniz
The modern binary system is often credited to Gottfried Wilhelm Leibniz, who saw a deeper, almost philosophical beauty in it. He associated binary’s simplicity — the concept of something \(\mathsf{1}\) emerging from nothing \(\mathsf{0}\) — with ideas of creation and metaphysics.
Extending Numbers in Binary
Just like in decimal, we build larger numbers by combining digits and assigning them place values. The key difference lies in the base:
- Decimal works in powers of \(\mathsf{10}\) like \(\mathsf{10^{-1}, 10^0, 10^1, 10^2, \dots}\).
- Binary works in powers of \(\mathsf{2}\) like \(\mathsf{2^{-1}, 2^0, 2^1, 2^2, \dots}\).
Because binary has only two digits, we quickly move into multi-digit numbers. Here’s how the first few look:
Every time we run out of combinations with the digits we have, we simply add a new digit to the left — just like decimal goes from \(\mathsf{9}\) to \(\mathsf{10}\), binary goes from \(\mathsf{1}\) to \(\mathsf{10}\).
Binary Numbers with Fractional Parts
Just like decimal numbers, binary numbers can include fractional parts. To separate the whole number portion from the fraction, we use a binary point — similar to a decimal point, but based on powers of \(\mathsf{2}\) rather than \(\mathsf{10}\).
Whole Number Side of the Binary Point
The digits to the left of the binary point represent the whole number. Their place values increase from right to left using powers of \(\mathsf{2}\):
\[\mathsf{2^0,2^1,2^2,2^3,\dots}\]
Here, the digit at position \(\mathsf{2^0}\) is the least significant bit of the integer part. As we move left, each bit becomes more significant.
Fractional Side of the Binary Point
The digits to the right of the binary point make up the fractional part. Their place values decrease from left to right using negative powers of \(\mathsf{2}\):
\[\mathsf{2^{-1},2^{-2},2^{-3},2^{-4},\dots}\]
The first bit after the binary point is actually the most significant bit of the fractional part, holding a value of \(\mathsf{2^{-1}=0.5}\). The next bit contributes \(\mathsf{2^{-2}=0.25}\), and so on.