If you've ever…

• Wondered by 0.1 + 0.2 equals 0.30000000000000004
• Were curious why you may get an integer errors when adding up large numbers, or
• Why you might get a small number when adding two big numbers

Then read on. In this multi part article series, you will learn the answer to all of these questions. I guarantee you that if you take the time to read these articles, you will be a much better coder. Knowledge of how a computer represents numbers will help you design more precise and stable applications.

Let's Begin!

Base 2 Number System

The base 10 number system, aka the decimal system, is the number system we use in our everyday life. As you all know in the decimal system each digit of a number can have a value ranging from 0 to 9. The places and positions of the numbers are based on powers of 10.

etc.	Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Ones
…	105	104	103	102	101	100
…	4	7	6	9	4	5

Each number position is 10 times the value to the right of it.

A computer on the other hand uses a different number system for representing numbers. Every piece of data stored in a computer system is represented in base 2. This is also known as a binary number. Each position in a binary number are based on powers of 2.

etc.	256	128	64	32	16	8	4	2	1
…	28	27	26	25	24	23	22	21	20
…	1	0	0	1	1	1	0	0	1

Each position in a binary number can have a value of either 0 or 1.

Here's an example of a binary number:

100011

The binary number above represents the decimal number 35. We'll get to how to convert from binary to decimal  in a bit, but let's quickly cover counting in binary.

Counting in Binary

Below is a table of numbers from 0 to 20 and their representation in binary. Can you see the pattern?

Number	20	19	18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1	0
Binary	10100	10011	10010	10001	10000	01111	01110	01101	01100	01011	01010	01001	01000	00111	00110	00101	00100	00011	00010	00001	00000

Counting in binary is very similar to how we count in decimal.  In decimal we start counting at 0 then count from 1 all the up to 9, then start at 0 again but add 1 to the digit on the left, resulting in the number 10.  Counting in binary is the exact same process, except we reach 10 much sooner.

Converting from Binary to Decimal

To convert a binary number to a decimal number, multiply each digit by its binary positional value and then all together. Here's an example to make things clearer:

100011

• The first digit on the left is 1. It's is in the 2⁵ position, making 1 * 2⁵ = 32
• The next digit is 0. It's in the 2⁴ position, meaning 0 * 2⁴ = 0
• The next digit is 0. It's in the 2³ position, meaning 0 * 2³ = 0
• The next digit is 0. It's in the 2² position, meaning 0 * 2² = 0
• The next digit is 1. It's in the 2¹ position, meaning 1 * 2¹ = 2
• The next digit it 1. It's in the 2⁰ position, meaning 1 * 2⁰ = 1

Adding these products together, we get 35.

Converting from Decimal to Binary

Converting from decimal to binary is accomplished by repeatedly dividing the decimal number by 2 until the quotient becomes 0. The remainder of the division at each iteration becomes a digit in the binary number.

Here's an example:

Converting 13 to binary

Division by 2	Quotient	Remainder
13/2	6	1
6/2	3	0
3/2	1	1
1/2	0	1

13 is 1101 in binary

That's it for now. Next article in the series, we will talk about positive and negative numbers in binary.