-
110110
-
1001000
-
111000
-
110111
EXPLANATION
Binary to decimal conversion result in base numbers
(111000)2 = (56)10
Binary System
The binary numeral system uses the number 2 as its base (radix). As a base-2 numeral system, it consists of only two numbers: 0 and 1.
While it has been applied in ancient Egypt, China and India for different purposes, the
binary system
has become the language of electronics and computers in the modern
world. This is the most efficient system to detect an electric signal’s
off (0) and on (1) state. It is also the basis for binary code that is
used to compose data in computer-based machines. Even the digital text
that you are reading right now consists of binary numbers.
Reading a binary number is easier than it looks: This is a positional
system; therefore, every digit in a binary number is raised to the
powers of 2, starting from the rightmost with 2
0. In the binary system, each binary digit refers to 1 bit.
How to Read a Binary Number
In order to convert binary to decimal, basic knowledge on how to read
a binary number might help. As mentioned above, in the positional
system of binary, each bit (binary digit) is a power of 2. This means
that every binary number could be represented as powers of 2, with the
rightmost one being in the position of 2
0
Example: The binary number (1010)
2 can also be written as follows:
(1 * 2
3) + (0 * 2
2) + (1 * 2
1) + (0 * 2
0)
Decimal System
The decimal numeral system is the most commonly used
and the standard system in daily life. It uses the number 10 as its
base (radix). Therefore, it has 10 symbols: The numbers from 0 to 9;
namely 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
As one of the oldest known numeral systems, the
decimal numeral system
has been used by many ancient civilizations. The difficulty of
representing very large numbers in the decimal system was overcome by
the Hindu–Arabic numeral system. The Hindu-Arabic numeral system gives
positions to the digits in a number and this method works by using
powers of the base 10; digits are raised to the n
th power, in accordance with their position.
For instance, take the number 2345.67 in the decimal system:
- The digit 5 is in the position of ones (100, which equals 1),
- 4 is in the position of tens (101)
- 3 is in the position of hundreds (102)
- 2 is in the position of thousands (103)
- Meanwhile, the digit 6 after the decimal point is in the tenths (1/10, which is 10-1) and 7 is in the hundredths (1/100, which is 10-2) position
- Thus, the number 2345.67 can also be represented as follows:
(2 * 103) + (3 * 102) + (4 * 101) + (6 * 10-1) + (7 * 10-2)
How to Convert Binary to Decimal
There are two methods to apply a binary to decimal conversion. The
first one uses positional representation of the binary, which is
described above. The second method is called double dabble and is used
for converting longer binary strings faster. It doesn’t use the
positions.
Method 1: Using Positions
- Step 1: Write down (1010)2 and determine the positions, namely the powers of 2 that the digit belongs to.
- Step 2: Represent the number in terms of its positions. (1 * 23) + (0 * 22) + (1 * 21) + (0 * 20)
- Step 3: (1 * 8) + (0 * 4) + (1 * 2) + (0 * 1) = 8 + 0 + 2 + 0 = 10
- Therefore, (1010)2 = (10)10
(Note that the digits 0 in the binary produced zero values in the decimal as well.)
Method 2: Double Dabble
Also called doubling, this method is actually an algorithm that can
be applied to convert from any given base to decimal. Double dabble
helps converting longer binary strings in your head and the only thing
to remember is ‘double the total and add the next digit’.
- Step 1: Write down the binary number. Starting from the left,
you will be doubling the previous total and adding the current digit. In
the first step the previous total is always 0 because you are just
starting. Therefore, double the total (0 * 2 = 0) and add the leftmost
digit.
- Step 2: Double the total and add the next leftmost digit.
- Step 3: Double the total and add the next leftmost digit. Repeat this until you run out of digits.
- Step 4: The result you get after adding the last digit to the previous doubled total is the decimal equivalent.
Now, let’s apply the double dabble method to same the binary number, (1010)
2
- Your previous total 0. Your leftmost digit is 1. Double the total and add the leftmost digit
(0 * 2) + 1 = 1
- Step 2: Double the previous total and add the next leftmost digit.
(1 * 2) + 0 = 2
- Step 3: Double the previous total and add the next leftmost digit.
(2 * 2) + 1 = 5
- Step 4: Double the previous total and add the next leftmost digit.
(5 * 2) + 0 = 10
This is where you run out of digits in this example. Therefore, (1010)
2 = (10)
10
Binary to decimal conversion examples
Example 1: (1110010)
2 = (114)
10
Method 1:
(0 * 2
0) + (1 * 2
1) + (0 * 2
2) + (0 * 2
3)
+ (1 * 2
4) + (1 * 2
5) + (1 * 2
6)
= (0 * 1) + (1 * 2) + (0 * 4) + (0 * 8) + (1 * 16) + (1 * 32) + (1 * 64)
= 0 + 2 + 0 + 0 + 16 + 32 + 64 = 114
Method 2:
0 (previous sum at starting point)
(0 + 1) * 2 = 2
2 + 1 = 3
3 * 2 =6
6 + 1 =7
7 * 2 = 14
14 + 0 =14
14 * 2 = 28
28 + 0 =28
28 * 2 = 56
56 + 1 = 57
57 * 2 = 114
Example 2: (11011)
2 = (27)
10
Method 1:
(0 * 2
0) + (1 * 2
1) + (0 * 2
2) + (1 * 2
3)
+ (1 * 2
4)
= (1 * 1) + (1 * 2) + (0 * 4) + (1 * 8) + (1 * 16)
= 1 + 2 + 0 + 8 + 16 = 27
Method 2:
(0 * 2) + 1 = 1
(1 * 2) + 1 = 3
(3 * 2) + 0 = 6
(6 * 2) + 1 = 13
(13 * 2) + 1 = 27
