Binary Made Easy
Binary numbers might seem mystifying at first glance, but this article breaks it down into familiar logic you use every day. By explaining the parallels between counting in tens and counting in twos, you'll be guided to understand how ones and zeros can represent anything!
Many people find binary intimidating, but it's simpler than you think. If you can count, then you can read binary. Today, we will break down this problem by reviewing how we understand numbers and then applying that same logic to binary.
Decimal System
In everyday life, when we count we typically use the decimal numbering system. In this system, a single digit is often referred to as a unit:
- 0 = no units
- 1 = one unit
- 2 = two units, etc.
When we reach ten units, we reset our units to zero and add another digit to the left and then continue incrementing the unit. This new digit to the left represents a multiple of 10, creating the "tens" column. Each column added to the left is another multiple of 10: so the pattern becomes units, tens, hundreds, thousands etc. The table below should be familiar:
| Thousands | Hundreds | Tens | Units | Name | Number |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | One | 1 |
| 0 | 0 | 1 | 0 | Ten | 10 |
| 0 | 1 | 0 | 0 | One Hundred | 100 |
| 1 | 0 | 0 | 0 | One Thousand | 1,000 |
| 2 | 3 | 4 | 5 | Two Thousand Three Hundred Forty-Five | 2,345 |
Recognising Binary
Decimal numbers can be described as "a sequence of digits ranging from 0 to 9". In the very same fashion, we can describe binary numbers as "a sequence of digits ranging from 0 to 1". The following examples are all binary numbers:
- 11001100
- 0001
- 0
- 1111
Binary System
The binary numbering system is a cousin to the decimal numbering system. It is used to represent the same numbers, they just look different.
We apply the same process to binary as we do with decimal but use only two digits (0 and 1) and each column is a multiple of 2 instead of 10. This means the pattern for our columns becomes units, twos, fours, eights, sixteens etc.
Take some time to look over this table. Pay close attention to the column names and compare them to the decimal table above. Can you see a pattern?
| Eight | Four | Two | Unit | Name | Binary Number |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | One | 0001 |
| 0 | 0 | 1 | 0 | Two | 0010 |
| 0 | 0 | 1 | 1 | Three | 0011 |
| 0 | 1 | 0 | 0 | Four | 0100 |
| 0 | 1 | 0 | 1 | Five | 0101 |
Converting Binary to Decimal
In my experience, manually converting a few binary numbers into decimal helps to solidify the concept. Let's use the binary number 101101 as an example.
First, draw the columns around the binary number (remember to use multiples of 2). I will use "one" for clarity instead of "unit".
| Thirty-Two | Sixteen | Eight | Four | Two | One |
|---|---|---|---|---|---|
| 1 | 0 | 1 | 1 | 0 | 1 |
Next, convert each digit based on its position.
| Thirty-Two | Sixteen | Eight | Four | Two | One |
|---|---|---|---|---|---|
| 32 | 0 | 8 | 4 | 0 | 1 |
Lastly, add them up:
Which means 101101 equals 45. Much easier once you know how it works, right?