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What is Binary Coded Decimal (BCD)?

One of the limitations of binary is its inability to store certain fractions accurately. For example, it is not possible to store 0.1 in binary (it’s an irrational number when that radix is used). This leads to rounding errors when using floating-point arithmetic, which can cause significant problems, especially when the arithmetic is performed within a loop, compounding the issue.

Binary Coded Decimal (BCD) solves this problem, among others, by encoding each denary digit in its own nibble. Consider the binary integer encoding for 204:

This value is completely different in BCD, where the priority is not to encode data as efficiently as possible, but represent each denary digit as a separate unit:

Of course, this is inefficient; a nibble can store 16 values, and BCD only uses each one for a value between 0 and 9. But the separation of digits provides some advantages:

  • No rounding errors: When digits are stored individually, they can be stored perfectly accurately, eliminating rounding errors.
  • Simpler electronics can be used: Some devices are very simple, and it is not worth including the circuitry required for them to interact with binary integers. Using BCD allows these devices to be created with much simpler electronics[1].

Common applications of BCD, predictably, include devices that benefit from one or both of these advantages:

  • Calculators: Using BCD avoids rounding errors. Also, the user inputs numeric values one digit at a time. Using BCD allows the calculator to support this input style without adding unnecessary complexity.
  • Alarm clocks: While it is very possible to store the current time, and the user’s chosen alarm time, as binary integers, the circuitry involved is quite complex. It is much simpler to store the times in BCD and only sound the alarm when all four digits in the current time match the four digits in the alarm time.
  • Simple embedded systems: Temperature alarms, air conditioners, microwaves, and more. These devices might not use BCD in reality (it depends on the choices made by the designers), but they are good candidates. For this reason, they might be featured in an exam question, so it’s worth bearing them in mind.

BCD has its disadvantages. The main issue is that it is a less efficient way of storing denary values, requiring roughly 20% more memory.

[1] This isn’t universally the case. General arithmetic using BCD requires 10-20% greater complexity when compared to using binary. Nonetheless, for simple devices, BCD is simpler.

Two’s complement: how to store negative integers in binary

Up to this point, we have only dealt with how to encode positive integers in binary. To encode negative integer values you must use two’s complement binary encoding. You are used to using eight bits to store any of 256 possible values ranging from 0 to 255. In two’s complement, you still use eight bits to store any of 256 possible values, but this time they range from -128 to 127. Two’s complement is an encoding method that is, at first, a little confusing. However, once you understand the process it is actually very simple. There are two methods. Both methods involve working with a value and flipping the sign (e.g. if you want the two’s complement binary for -56, write out the binary for 56 and then perform the calculations required to flip the sign, giving you -56). Note that the same method can be also used to convert a negative value into a positive one. In our examples below, we will convert 116 into -116 using two’s complement encoding.

Method 1

This method is simple. Firstly, write out the binary value:

Note that the Most Significant Bit (MSB) is used to represent -128, not 128. You can use this to check your answers if you’re unsure in the exam.

Next, flip every bit:

Finally, using binary addition[1], add 1 to the value:

You now have the two’s complement binary encoding for -116.

Method 2

Start from the Least Significant Bit (LSB), moving left until you find a 1:

 Ignore the first ‘1’:

Flip every bit that follows:

You now have your answer.

A few important notes

Firstly, two’s complement is used to encode positive values as well as negative values. If you’re asked to give the two’s complement for a positive value, it’s exactly the same as it would be if it were encoded as a standard binary integer. Read the question carefully.

Two’s complement values don’t have the same range as their positive binary integer equivalents. You can’t store 128 in an 8-bit two’s complement value. You’ll need at least nine bits to store it. Be careful.

All two’s complement values where the MSB is ‘1’, are negative. Always. The opposite is also true.

Finally, if you ever need to pad bits (i.e. add more bits to the left of a binary value to make it a certain length, such as a byte), just as you’d use zeroes to pad a positive value (e.g. 0011 -> 00000011), you must use ones to pad a negative value (e.g. 1011 -> 11111011).

[1] Put simply, binary addition is very similar to denary addition, including ‘carrying the one’. The only difference is that digits can only range from 0-1. Binary addition depends on some very simple rules: 0 + 0 = 0; 0 + 1 = 1; 1 + 0 = 1; 1 + 1 = 0 carry 1; 1 + 1 with a carried 1 = 1 carry 1. You can find plenty of online tutorials for this if you wish to understand it further.

How to convert binary and denary to hexadecimal

Hexadecimal uses the digits 0-9 and the letters A-F, giving sixteen possible values per digit. The values are assigned like so:

Denary0123456789101112131415
Hex0123456789ABCDEF


You must be familiar with this table to be able to understand hexadecimal questions and the number system as a whole.

  1. Split the binary value into nibbles (four-bit values).
  2. Write the numbers over every bit, treating every nibble as a separate binary value (this means you write 1, 2, 4, and 8 over the digits in every nibble, starting at the right).
  3. Convert every nibble to denary. Each nibble will give you a value between 0 and 15.
  4. Convert each denary value to its hexadecimal equivalent. Use the table above if you don’t remember the equivalent values
  5. You now have the hexadecimal equivalent of the binary.

This process is shown in the diagram below:

Other conversions are also possible using similar techniques.

Converting from hexadecimal to binary

  1. Write the denary value for each hexadecimal digit.
  2. Convert each denary value into a nibble.
  3. You now have the binary equivalent of the hexadecimal value. Make sure you don’t write any spaces between the nibbles for your final answer. 

Converting from denary to hexadecimal

  1. Convert the denary value to its binary equivalent.
  2. Follow the steps for converting binary to hexadecimal.

Converting from hexadecimal to denary

  1. Convert the hexadecimal value to its binary equivalent.
  2. Follow the steps for converting binary to denary, which can be found in this post.

How to convert between binary and denary

Converting from binary to denary, and from denary to binary, is a quick and easy process that only requires very basic mathematical knowledge and skills. There are a few ways to these conversions; the techniques below are the simplest and most reliable for most people.

Converting from binary to denary

The process is as follows:

  1. Write a small 1 above the rightmost bit.
  2. Moving from right to left, write numbers above each bit, doubling the value every time. It should look like this:

    If the binary value is more than eight digits in length, keep writing the numbers, doubling the value every time (i.e. 256, 128, 512, etc.); the exam board could ask you to work with numbers as long as 16 bits in length.
  3. Add up the numbers above every bit that’s a ‘1’ (in this case, 64 + 32 + 4 + 2 = 102). The total is the denary value.

Converting from denary to binary

The process is essentially the reverse:

  1. Write the numbers above every bit, just as you did before.
  2. Find the bit with the number value that’s less than or equal to the denary value.
  3. Write a ‘1’ in that column and subtract that value from your denary.
  4. Repeat this process until your denary is 0.
  5. Fill all the remaining bits with ‘0’s.

What is binary?

There are several different number systems, each with a different base (sometimes called the radix). The base of a number system tells us how many different possible values a single digit can take. The number system we’re most used to is denary[1], which is base 10 (because it has ten possible values per digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9). Binary is a base 2 number system, giving it two possible values per digit (0 and 1).

In denary, every digit has a value 10 times greater than the digit to its right:

To get the overall value, we simply multiply the value of the column by the value of the digit:

(9 x 100) + (4 x 10) + (2 x 1) = 941

Binary works using exactly the same principles, except every column is worth double the column to its right:

Every digit in binary is called a bit. To convert between binary and denary integers, we need to follow the relevant process.

[1] Another common term is decimal, which is also correct.