# Why is Binary Used in Electronics?

## The Binary System

The binary numbering system, is the basis for storage, transfer and manipulation of data in computer systems and digital electronic devices. This system uses base 2 rather than base 10 which we are familiar with for counting in everyday life. This hub explains the basics and takes some of the mystery out of why the binary system is used.

## What is Decimal and Why Do We Use It?

The decimal, base 10 or denary numbering system is what we are familiar with in everyday life. It uses 10 digits. So you count 0,1,2,3,4,5,6,7,8,9 and when you get to 9, there is no symbol for the next number ten, so ten is represented by the digit 1 followed by 0 or "10", which really means "one ten and no units". Similarly when you get to 99, one hundred is represented by three digits 1, 0 and 0 i.e "100 which is interpreted as "one hundred, no tens and no units". So basically numbers are represented by a series of symbols in the units, tens, hundreds, thousands place etc. So for instance 134 means "one hundred, three tens and four units. The decimal system probably arose because we have 10 fingers on our hands which could be used for counting.

## What is Binary and How Does it Work?

The binary system used by computers is based on 2 symbols; 0 and 1. So you count 0, 1, there is no symbol for 2, so 2 is represented by 10 or "one 2 and no units". In the same way that there is a units, tens, hundreds, thousands place in the decimal system, in the binary system there is a units, twos, fours, eights, sixteens place etc. in the binary system. So the binary and decimal equivalents are as below:

00000000 0

00000001 1

00000010 2

00000011 3

00000100 4

00000101 5

00000110 6

00000111 7 and so on

## Why Do Computers Use Binary?

*"A single switch can be on or off, enabling the storage of 1 bit of information. Switches can be grouped together to store larger numbers. **This is the key reason why binary is used in digital systems."*

### Binary in Digital Computers and Electronic Devices

Numbers can be encoded in binary format and stored using switches. The digital technology which uses this system could be a computer, calculator, digital TV decoder box, cell phone, burglar alarm, watch etc. Values are stored in binary format in memory, which is basically a bunch of electronic on/off switches.

Imagine if you had a bank of 8 rocker switches just like in the image below. Each switch could represent 1 or 0 depending on whether it is turned on or off. So you think of a number and set the switches on or off to "store" the binary value of this number. If someone else then looked at the switches, they could "read" the number. In a computer, switches are implemented using transistors. The smallest memory configuration is the bit, which can be implemented with one switch. If 8 switches are added together you get a byte. The digital hardware is able to set the switches on and off (i.e. write data) and also read the state of the switches. In the conceptual image below, there are 8 switches and 2 to the power of 8 = 256 *permutations* or arrangements depending on whether a switch is on or off. If on represents 1 and off represents 0, the group of switches can represent any of the following values.

00000000 0 decimal

00000001 1 decimal

00000010 2 decimal

00000011 3 decimal

00000100 4 decimal

.

.

.

.

11111110 254 decimal

11111111 255 decimal

In an electronic device or computer, because of micro-miniaturization, billions of switches can be incorporated onto integrated circuits (IC), potentially allowing the storage and manipulation of huge amounts of information.

## 8 Bit "Memory" Made From a Gang of Rocker Switches

## Floating Point, ASCII Code, Machine Code and Storage of Analog Data

Now in reality, an electronic system can be more complex. Numbers with decimal places are stored using a system called floating point and memory locations may be 32 or 64 bits wide.

Several types of data encoding systems are used by digital technology for representing data which is not specifically numbers. The data could be names and addresses in a database, temperature levels measured in a room, control data for an industrial robot or low level machine code which tells a microprocessor what to do.

A machine such as a computer, digital camera,, scanner etc cannot directly store non-numerical data from the real world in binary format. This type of data could be:

- A person's name
- Temperature in a room
- An image from a digital camera or scanner.
- Audio

The data must be encoded. In the case of a name or other text, the text is broken down into individual letters, numbers and other non alphanumeric characters (e,g, &*£$# etc) and a coding system called ASCII represents each character by a number from 0 to 127. This data is then stored in binary format as one or more bytes in memory, each byte being made up of individual bits, and each bit implemented using transistors. Similarly a voltage level from a temperature sensor is an analog signal and has to be converted to a number by a device called an analog to digital converter (ADC). These devices can have various resolutions and for a 16 bit converter, the signal level is represented by a number from 0 to 65535. A scanned image, digital photo or image created in a drawing package is stored as individual pixels and a byte of data is used for the red, green and blue intensity levels of each pixel.

Not only are values or data stored in memory, but also the instructions which tell the microprocessor what to do. These instructions are called machine code. When a software program is written in a high level language such as BASIC, Java or "C", another program called a compiler breaks the program down into a set of basic instructions called machine code. Each machine code number has a unique function which is understood by the microprocessor. At this low level, instructions are basic arithmetic functions such as add, subtract and multiply.

## Analog and Digital Signals

## ASCII Code Table

## Convert Decimal to Binary Using the Remainder Method

If you don't have a calculator, you can easily convert a decimal number to binary. Successively divide the decimal number by 2 until the result is 0, noting each remainder. Then read the list of remainders from the bottom up.

The MSB or "most significant bit" is the leftmost bit in the result.

The LSB or "least significant bit" is the rightmost bit in the result.

## Converting From Binary to Decimal

To convert from binary to decimal, multiply each digit in the binary number by successive powers of 2, and then add the results together

e.g. What is 1011010 in decimal?

Writing the binary number in a column with the MSB or leftmost bit at the top:

1 x 2^{6} = 64

0 x 2^{5} = 0

1 x 2^{4} = 16

1 x 2^{3} = 8

0 x 2^{2} = 0

1 x 2^{1} = 2

0 x 2^{0} = 0 (2^{0} = 1)

So 1011010 = 64 + 16 + 8 + 2 = 90 decimal

## Online Binary - Decimal - Hex Converter

This online converter lets you enter a binary, decimal or hexadecimal number which is instantly converted to the two other base systems:

## George Boole and Boolean Algebra

Boolean algebra, developed by the British mathematician George Boole in the 19th century, is a branch of mathematics which deals with variables which can only have one of two states, *true* or *false*. In the 1930s Boole's work was discovered by the mathematician and engineer Claude Shannon, who realized it could be used to simplify the design of telephone switching circuits. These circuits originally used relays which could be either on or off, and the desired output state of the system, depending on the combination of states of the inputs, could be described by a Boolean algebraic expression. Boolean algebra rules could then be used to simplify the expression, resulting in a reduction of the number of relays required to implement a switching circuit. Eventually Boolean algebra was applied to the design of digital electronic circuitry as we will see below.

## Digital Logic Gates - AND, OR and NOT

A digital state, ie high/low or 1/0 can be stored in a 1 bit cell in memory, but what if that data has to be processed? The most basic processing element in a digital electronic circuit or computer is a *gate*. A gate takes two or more digital signals and generates an output. There are three types of gates, AND ,OR and NOT (INVERT). In their simplest form, small groups of gates are available on a single IC. However a complex combinational logical function can be implemented using a *Programmable Logic Array *(PLA) and more sophisticated devices such as microprocessors are composed of millions of gates and memory storage cells.

For an AND gate, the output is true or high only when both inputs are true.

For an OR gate, the output is high if either or both inputs are true.

For a NOT gate or inverter, the output is the opposite state to the input.

Boolean algebraic expressions can be used to express what the output signal of a circuit should be, depending on the combination of inputs. The main operations in Boolean algebra are *and*, *or* and *not*. During a design process, the required value of an output for all the various permutations of input states can be tabulated in a *truth table. *The value '1' in the truth table means an input/output is true or high. The value '0' means the input/output is false or low. Once a truth table is created, a Boolean expression can be written for the output, simplified and implemented using a collection of logic gates.

*So a typical Boolean expression with three independent variables A,B and C and one dependent variable D would be:*

## Y = A.B + C

This is read as "Y = (A and B ) or C"

## Comments

You explained this very well. Many years ago I experimented on programming in machine language - what a hassle! It didn't take long to move to something higher.

It is a very good article that explains things very well. There is a simpler albiet less refined unrefined method of conversion. Decimal 27 to binary:

32 16 8 4 2 1

0 1 1 0 1 1 = 16+8+2+1 = 27

Binary 1011010 to decimal:

64 32 16 8 4 2 1

1 0 1 1 0 1 0 = 64+16+8+2 = 90 Decimal

5A Hexidecimal = 132 Octal

Crude but effective.