What is binary number system?

Some of you must be familiar with this number system, while others may be surprised to see this calculation. Yes in decimal number system, 1 + 1 = 2 but in digital number system 1 + 1 = 10. It’s not read as “Ten” rather it is read as “One Zero”. The decimal number system which we study in mathematics contains basic numbers 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. All other numbers in decimal number system, are combination of any of these numbers only. But as we saw in previous topic where we compare analog and digital system, we said that digital system works on only two states i.e. ZERO and ONE. Hence the number system in digital also contains only ‘0’ and ‘1’ and all other numbers are only a combination of 0 and 1. This number system is called Binary Number System.

As the digital number system contains “Two” numbers only, hence it is called a Binary number system. “Bi” indicates number system with two basic values. These two binary digits i.e. ‘1’ and ‘0’ are also called “Bit”. The word “Bit” is the combination of Binary + Digit. So, in digital number system ‘1’ is one Bit and ‘0’ is another bit. All digital communications happens based on these bits, may it be communication over wire or wireless.

When you press any key on your computer keyboard, the digital data from keyboard goes to computer over the wire connected to it. When you are browsing any website over internet the digital data may be going over wired broadband or wireless Wi-Fi, but all these communications are in the form of ONLY ‘0’ and ‘1’.

A single bit is denoted as small character ‘b’. So, whenever we talk about speed of data transfer over wired or wireless media, it’s “Kbps” or “Mbps” i.e. Kilo bits per second and Megabits per second. 1Kilobit = 10³bits and 1Megabit = 10⁶ bits.

All the characters in the digital computers are a combination of 8bits, like for letter “a” it is “0110 0001” and for letter “A” it is “0100 0001”. Similarly we have 8bit combination for all other characters. This basic combination of 8bits is called a “Byte” i.e. 1byte = 8bits. Actually we don’t have space in between four digits, but for easy representation we show in this form.

Now let us see how all other numbers are denoted using bits. When a number is represented in a combination of 8bits, every number has its place value as we have in decimal number system. As this is a binary number the place value is decided by “2ⁿ” where ‘n’ is the place value staring from right and moving towards left. First place value at extreme right is ‘0’, then ‘1’, ‘2’ and so on till we reach extreme left.

Now the process to find out the decimal number from binary digits is simply to multiply each digit with its place value and then add all the numbers.

For example if we have a four digit number as “0001” its decimal value will be (8X0) + (4X0) + (2X0) + (1X1) = 0 + 0 + 0 + 1 = 1.

Let’s take another example of binary number – “0101”, its decimal equivalent will be (8X0) + (4X1) + (2X0) + (1X1) = 0 + 4 + 0 + 1 = 5.

Similarly let’s take another example of 8bit number like – 0110 1010, based on above method its decimal equivalent is (128X0) + (64X1) + (32X1) + (16X0) + (8X1) + (4X0) + (2X1) + (0X0) = 0 + 64 + 32 + 0 + 8 + 0 + 2 + 0 = 106.

We have a standard formula that tells us how many unique combinations of bits we can make if we know the number of bits available with us. The formula is 2ⁿ, where ‘n’ is the number of bits. So, if we have two bits with us we can have 2² i.e. 4 different combinations. Similarly if we have three bits, we can have 2³ i.e. 8 different combinations. Examples are shown in the image with two bits and three bits.

This is how a binary number represent all decimal numbers. 

In digital system one more notation is used to show complementary number i.e. complementary of ‘1’ is ‘0’ and complementary of ‘0’ is ‘1’. This is denoted in symbol by placing a horizontal line on top of number as shown in picture.

So, any decimal number can be converted into binary number and for that we need to divide the number by ‘2’ and keep noting down the remainders. Every time the remainder will be either ‘0’ or ‘1’ only. Finally write the remainder numbers in sequence starting from bottom to top, to get its binary equivalent. As shown in this example, we have taken decimal number ‘25’ and we are finding the binary equivalent for it.

Step-1: 25 divided by 2, quotient is 12 and remainder is ‘1’.

Step-2: 12 divided by 2, quotient is 6 and remainder is ‘0’.

Step-3: 6 divided by 2, quotient is 3 and remainder is ‘0’.

Step-4: 3 divided by 2, quotient is 2 and remainder is ‘1’.

Step-5: Start writing the digits from last quotient and move toward top as shown by arrow, you will get “11001” and to show in a byte format you can add additional zeros in beginning to make it 8bit like “0001 1001”

Addition of binary numbers is done as shown in this table. As we mentioned in the first line 1 + 1 = 10 i.e. 1 + 1 = 0 with carry ‘1’ which is placed before zero that’s the reason it is written as ‘1’ and ‘0’. It should not be read as “Ten”, as in digital number system we have only two bits i.e. ‘1’ and ‘0’.

That’s the basic of binary number system, next we will see about digital gates which are the building blocks of digital circuits.

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