Blog Post 5

\(1\) \(\underline{\text{Binary Notation}}\)

We define binary numbers as the powers of two that lay the foundation for the additive building blocks of positive integers. Note that the word binary comes from “Bi” meaning two. In this system, integers are expressed in terms of only \(0\)s and \(1\)s. The values that represent each integer are calculated by finding the sum of the powers of two that make up the given number. We pull out the amount of times that each power of two occurs. For example, the decimal number ten is written as “\(1010\)” because it is \(\underline{1}\) \(\cdot\) \(2^3+ \underline{0} \) \(\cdot\) \(2^2 + \underline{1} \cdot 2^1 + \underline{0} \cdot 2^0 \). Notice that this starts with the largest power of two. We read “\(1010\)” as “one-zero-one-zero” as opposed to one thousand and ten. The binary representation of the first few natural numbers are shown in the table below.

Pictured (above) is the binary representation of the numbers 1-8.

You can see that the binary representation of the decimal number \(1\) is “\(1\).” A single binary digit is called a bit. Notice that the word “bit” comes from \(\underline{\textit{b}}\)inary dig\(\underline{\textit{it}}\).

When writing binary numbers more formally, we tend to use the subscript \(2\). Therefore, we would write the decimal number \(5\) whose binary notation is “\(101\)” as \(101_2\). This is so that there is less confusion between decimal numbers and binary numbers.

Today, computers use binary numbers because they are simple to work with. Numbers can be encoded in binary and stored using the idea of switches. Each switch represents either a \(1\) or a \(0\) depending on whether the switch is on or off. Therefore, by looking at a switch, or set of switches, you could tell what number is being displayed. The storage of large amounts of information can be allowed by having several switches. “The digital technology which uses this system could be a computer, calculator, digital TV decoder box, cell phone, burglar alarm, watch etc.” (Brennan, 2016).

When binary numbers were invented by Gottfried Wilhelm Leibniz, it was thought that this system could be used as a primitive calculating machine. Nowadays, binary numbers are at the foundation of all modern computing systems.