Digital Electronics

Number System

NUMBER SYSTEM

Can Computers understand Decimal numbers???

🌾Since humans can understand decimal number system, in earlier days scientists were more interested in making machines that can directly operate on decimal number system. Few examples of ‘decimal computers’ are as follows,

Decimal Computers

🌾Decimal computers are machines that use decimal numbers as a base for programming, calculations, storage, etc.,

Mechanical machines:

  • In $1883$, Charles Babbage theorized an ‘Analytical engine’ – a mechanical computer which employs decimal number system for its programming and calculations.
  • In $1909$, Ludgate theorized an ‘Analytical engine’. It also employs decimal number system.

🌾As in the above figure, in mechanical machines, gears with $10$ teeth are used to implement decimal numbers.

Positions or states:

  • Tooth $1$ is used to represent the decimal number $0$. Hence it is position $1$ or state $1$.
  • Tooth $2$ is used to represent the decimal number $1$. Hence it is position $2$ or state $2$.
  • Tooth $3$ is used to represent the decimal number $2$. Hence it is position $3$ or state $3$
  • and so on.

Stable Positions or states:

After rotating the gear to a particular position, If the gear does not go back to its previous position (until it is disturbed), then that position is said to be a stable one. 

Hence, we can say, gears with $10$ teeth have $10$ stable positions or states. These ten teeth gears serve as basic building block for mechanical decimal computers.

Gears can have $10$ stable positions or states. 

Hence, Gears can be used to implement Decimal number system in mechanical machines.

Electro-Mechanical machines:

  • In $1920$, Leonardo Torres Quevedo theorized an ‘Analytical machine’ – an electro-mechanical computer. It employs decimal number system.
  • In $1944$, Howard Aiken made an ‘Analytical machine’ named Harvard Mark I – an electro-mechanical computer. It also employs decimal number system.

🌾Relays are the basic building block of any electrical machines. Relays can only act as a switch, it has only two positions or states {normally closed (OFF) and normally open (ON)}.

Normally open (ON): When the coil is energized, the electromagnet will become a magnet. This magnet will attract the top iron, which will make this relay ‘ON’ or normally open.

Normally closed (OFF): When the electromagnet iron is not energized, the top iron will go to its original position, which is ‘OFF’ or normally closed (as shown in the above figure).

From this we can say, relays can have only two positions or states.  Among them, OFF position is stable and ON position is astable.

Relays can only have $2$ positions or states.

Is it enough to represent decimal numbers???

🌾To represent decimal numbers, we need $10$ stable positions or states, hence relay was combined with gear using clutch to make a ten position relay, as given below

Relay + Gear = Electro + mechanical

🌾See the arm in the above picture, it can be placed at, 

  • position $0$ to represent the decimal number $0$
  • position $1$ to represent the decimal number $1$
  • position $2$ to represent the decimal number $2$ (as shown in the figure)
  • position $3$ to represent the decimal number $3$
  • and so on.

Hence, we can say, ten position relay have $10$ stable positions or states. This ten position relay serves as a basic building block for electro-mechanical decimal computers.

Ten position relay can have $10$ stable positions or states

Hence, Ten position relay can be used to implement Decimal number system in electro-mechanical machines.

Electronic machines:

  • In $1945$, ENIAC (Electronic Numerical Integrator and Calculator) – a purely electronic machine has been built using vacuum tubes. It employs decimal number system!!!

How a decimal number is represented in ENIAC???

🌾Few lines from the book, “Giant Brains or Machine that think” by Edmund C. Berkeley, $1949$

Basically, a number is represented in ENIAC by an arrangement of ON and OFF electronic tube elements in pairs, called flip-flops. There is one flip-flop enclosed in a single tube (type $6SN7$) for each value $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ for each of the ten digits stored in an accumulator.

From the above lines, 
  • ENIAC used flip-flops (made up of vacuum tubes), an electronic component which has two stable positions or states. $$ON-OFF$$ 
  • But to implement decimal numbers in a machine, we need $10$ stable positions or states.
  • Hence, ENIAC engineers used ten position ring counter ($10$ flip-flops) to represent a single digit. (i.e.) $10$ bits to represent a single decimal digit.
  • Ten position ring counter: Imagine, one flip-flop for each position in mechanical gear.

Each circle represents a flip-flop

How flip-flops are used to represent decimal number??

  • Flip-flop $1$ in position $1$ (rightmost) is used to represent the presence of the number $0$.
  • Flip-flop $2$ in position $2$  is used to represent the presence of the number $1$.
  • Flip-flop $3$ in position $3$ is used to represent the presence of the number $2$.
  • and so on
For example, in the above figure, flip-flop $4$ is ON (shaded in yellow) or $’1’$, it represents the presence of number $3$ in that digit.
$0$ – $0000000001$
$1$ – $0000000010$
$2$ – $0000000100$
$3$ – $0000001000$
$4$ – $0000010000$
$5$ – $0000100000$
$6$ – $0001000000$
$7$ – $0010000000$
$8$ – $0100000000$
$9$ – $1000000000$

 

The above image shows ten digit – ten position ring counter output. The number in the above image is $+0000075000$

ENIAC used around $18,000$ vacuum tubes !!!

Summarizing, 

  • $10$ states: To make a machine, to understand decimal numbers, we need components that can have $10$ positions or states.
  • Gears: Gears can have $10$ positions or states, hence mechanical machines can be made to understand Decimal numbers.
  • Ten position relay: Ten position relay (Relay combined with gear using clutch) can have $10$ positions or states, hence electro-mechanical machines can be made to understand Decimal numbers.
  • ENIAC: Using ten position ring counter ($10$ flip-flops for single digit), Decimal numbers were represented in ENIAC.

Now answering the question, “Can computers understand Decimal numbers??”

Yes

Mechanical machines, electro-mechanical machines and some old electronic machines were made to understand Decimal numbers.

But modern computers only uses binary number system. Why???

References:

  • ‘Giant Brains or Machines that think’ by Edmund C Berkeley.

  • Wikipedia – https://en.wikipedia.org/wiki/Mechanical_computer

  • Youtube video: Computer History: 1946 ENIAC Computer History 
  • Charles Babbage Analytical image courtesy – https://www.computerhistory.org/babbage/engines/