Binary Number System – Lucid Explanation of Conversion From and to Decimal Number System – Examples

The base-10 number system or decimal number system is the most popular system used by humans throughout the world.

But, computers work internally with only two symbols, because of the straightforward implementation in digital electronic circuitry using logic gates.

Thus, the base-2 number system or Binary Number System is the basis for digital computers.

It is used to perform integer arithmetic in almost all digital computers.

The two base symbols or digits used in Binary number system are 0 (called zero) and 1 (called one).

We are already familiar with these symbols or digits in decimal number system.

Let us learn how to write numbers using the binary number system.

This system is analogous to the decimal number system in following the place value rule.

There, value of the place becomes ten times, as we move one place to the left, and here it becomes two times.

Place value rules in Binary number System :

The value of the right extreme place is one (1) or unity.

Value of the place increases as it moves to the left.

Value of the place becomes two times, as we move one place to the left.

So, the value of the place second from right is two times one and is equal to two.

The value of the place third from right is two times two and is equal to four.

The value of the place, fourth from right is two times four and is equal to eight.

The value of the place, fifth from right is two times eight and is equal to sixteen.

Thus, the next place values are thirty two, sixty four, one hundred twenty eight and so on.

I Conversion of base-two numerals into base-ten numerals :

The following examples will make the process clear.

Example I(1) :

Find the value of the binary numeral 1001, in the Decimal number system.

Solution :

In the given Binary Numeral,

units’ place (Extreme right place) has 1.

Twos’ place (Second place from right) has 0.

Fours’ place (Third place from right) has 0.

Eights’ place (Fourth place from right) has 1.

The value of the given binary numeral (1001) in Decimal Number System

= 1 ones + 0 twos + 0 fours + 1 eights

= 1 + 0 + 0 + 8 = 9. Ans.

Example I(2) :

Write the binary numeral 10010, in the Decimal number system.

Solution :

Binary Digit :   0  1  0  0  1

Place Value :  1  2  4  8  16

The Binary numeral 10010, in Decimal Number System

= 0(1) + 1(2) + 0(4) + 0(8) + 1(16) 

= 0 + 2 + 0 + 0 + 16

= 18. Ans.

Example I(3) :

Write the binary numeral 1110011, in the Decimal number system.

Solution :

Binary Digit :   1   1   0   0   1   1   1  

Place Value :  1   2   4   8   16  32  64


The Binary numeral 1110011, in Decimal Number System

= 1(1) + 1(2) + 0(4) + 0(8) + 1(16) + 1(32) + 1(64) 

= 1 + 2 + 0 + 0 + 16 + 32 + 64

= 115. Ans.

II Conversion of base-ten numerals into base-two numerals :

We use division method.

We successively divide by 2 and take the REMAINDER 0 or 1 in successive places starting from units’ place.

We continue the process till the quotient is 0.

The following examples will make the process clear.

Example II(1) :

Write the Decimal Numeral 36, in the Binary Number System.

Solution :

2 | 36


2 | 18  –  0     Units’ place


2 |  9  –  0     Twos’ place


2 |  4  –  1     Fours’ place


2 |  2  –  0     Eights’ place


2 |  1  –  0     Sixteens’ place


# |  0  –  1     Thirty two’s place

In the above presentation,

first column is having twos with which we are dividing.

Second column is the quotient obtained by dividing with 2.

# indicates the end of operation when the quotient is 0.

Third column (after ‘-‘) is the remainder (0 or 1) obtained which is the digit taken in successive places starting from units’ place.

Thus, The Decimal Numeral, 36 = 100100 in Binary Number System.

Example II(2) :

Write the Decimal Numeral 101, in the Binary Number System.

Solution :

2 | 101


2 |  50  –  1     Units’ place


2 |  25  –  0     Twos’ place


2 |  12  –  1     Fours’ place


2 |   6  –  0     Eights’ place


2 |   3  –  0     Sixteens’ place


2 |   1  –  1     Thirty two’s place


# |   0  –  1     Sixty four’s place

Thus, The Decimal Numeral, 101 = 1100101 in Binary Number System.

Example II(3) :

Write the Decimal Numeral, 1227 in the Binary Number System.

Solution :

2 | 1227


2 |  613  –  1    units’ place


2 |  306  –  1    Twos’ place


2 |  153  –  0    Fours’ place


2 |   76  –  1    Eights’ place


2 |   38  –  0    Sixteens’ place


2 |   19  –  0    Thirty twos’ place   


2 |    9  –  1    Sixty Fours’ place


2 |    4  –  1    One hundred twenty eights’ place


2 |    2  –  0    Two hundred fifty six’ place


2 |    1  –  0    Five Hundred twelve’s place


# |    0  –  1    One thousand twenty four’s place

Thus, The Decimal Numeral, 1227 = 10011001011 in Binary Number System.

What Is the Binary Arithmetic System?

In our daily life we use the decimal arithmetic system for our calculations. On the other hand computers do not use the decimal system but the binary one. If someone wants to easily understand the binary system it is good to start realizing some meanings for the decimal system and then apply them to clarify the binary system. This article is going to shortly, clearly and effortlessly explain what the binary system is.

In the decimal system we can construct every decimal number using 10 unique digits which are the following: 0,1,2,3,4,5,6,7,8,9. This is the reason we say that the base of the decimal system is 10. So for instance, the decimal number 1034 consists of the digits 1, 0, 3 and 4. By getting digits from 0 to 9 we can construct every string of a decimal number. The digits might be repeated or not. For example the number 1041 is a valid decimal number even though the digit 1 repeats twice.

Each decimal number has 2 major characteristics. The first one is the digits of which it consists of and the second one is the place of the digits. What do we mean with the term place? Let’s take as an example the number 123. You see that the digit 3 defines units, the digit 2 defines decades (10 times the units) and the digit 1 defines hundreds (100 times the units). So we can say that the most significant (the most weighted factor) digit is the left one and the least significant digit is the right one. In our case the most significant is the digit 1 and the least significant is the digit 3.

Now, we can easily explain and understand the binary system. In relation to the decimal system the binary system consists of numbers made by unique digits 0 and 1. Since these digits are only 2 this is the reason we call it binary system. The base of the binary system is 2. By getting digits from 0 to 1 we can construct every string of a binary number. For example the number 101010101 is a valid binary number. The number 1012010 is not valid because it uses the digit 2 which is not valid binary digit. The most significant digit is the left one and the least significant digit is the right one exactly the same as the decimal system. I would like to notice that computers and in general electronic devices use this system because they can get one of two unique states every moment, to have voltage (or current) or not.

So far we have clarified some meanings of the decimal system and applied them to the binary system aiming to easily understand it. I hope you enjoyed this article.

Binary Numbers and the Lightning Internet

Have you ever wondered how the Internet is organized? Imagine billions of devices sending and receiving information between each other. Whether computers communicate from within the same room or across the other side of the world doesn’t seem to matter in terms of finding each other. So how do all the Internet connected devices find each other within milliseconds of a request?

When it comes to the digital world, the binary number system is used to provide logical mathematical addresses. But binary numbers were not selected because they are an ideal numeric system in computers, in fact, binary is the most inefficient number system because it is limited to the use of just two digits – 0 and 1. But it just so happens that when it comes to electrical pulses between 0 and 5 Volts we are limited to on and off – and these voltages are easily translated to binary 1 and binary 0.

So it almost seems counter intuitive to say that billions of devices are finding and sending information between each other quickly and efficiently using the most inefficient number system imaginable. But we know that this is the case because we use and experience the Internet every day.

The unspoken hero behind this lightning locating is the addressing that drives the Internet. It is brilliantly constructed into a hierarchical structure much like the way we narrow down our own residence by Country, State, City, Street and Street number – the Internet uses an ingenious mechanism called masking in order to group large numbers of computers into a single statement.

Masking (or network masks) work by identifying large address blocks that can be further divided into smaller sections. Say for example that I assign all computers in the first floor of the building with an identifying number beginning with ‘1’ and all computers on the second floor with an identifying number beginning with ‘2’. No matter what sequence of numbers the floor administrators assigned to each computer within the floor we would always be able to identify the first floor and second floor computers very quickly by just looking at the first digit. This is a very fast and efficient process.

Network Masks on the Internet work in the same way. The power is in the flexibility. One can assign a very general mask – much like our previous first floor/second floor computer example – or a more specific mask in order to pinpoint a specific computer within a building.

So next time you access your favourite website without knowing whether you are accessing web servers close to you or on the other side of the world, take a moment to ponder the hierarchical brilliance that are the binary network masks and how this concept made possible a superhighway with millions of simultaneous visitors anywhere in the world using just ones and zeros.