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 Binary Arithmetic
Written by Mike James
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Binary Arithmetic
Choosing the right base

A little while after Shannon worked out that binary arithmetic was very easy to implement, the same idea was rediscovered by George Stibitz. He was musing on the problems of using relays in telephone circuits when he suddenly realised that he could use the open/closed positions as 0/1 and do arithmetic.

He quickly worked out a half adder and built it on his kitchen table using light bulbs to show the results. In 1940 he demonstrated a simple binary relay computer and in the audience was John Mauchly who went on to build, with John Eckert, the first electronic computer - ENIAC.

The big shock is that ENIAC, one of the first electronic computers, was a decimal computer not a binary computer.

Part of the reason was that the electro-mechanical computers that came before it were decimal but it must also have been that the advantages of binary just weren’t that obvious - even though this seems incredible.

The only exception to the rule that early computers were decimal was Conrad Zuse’s Z1 built in 1936. It also seems that he realised that binary was the way to build a computer in 1935, which gives him a strong claim to have thought of the idea before Shannon or Stibitz. Switching to binary was such a simplification that Zuse was able to build complete computers on his kitchen table.

Representing numbers in binary makes the connection not only with electro-mechanical switching circuits but with more advanced electronics. Once you settle on using binary it becomes very easy to find physical systems that have two states that can be used to represent 0 and 1. This makes creating storage and memory devices very simple.

You can use everything from holes punched in paper tape or cards to the direction of magnetisation to represent numbers in binary. However, there is a cost-efficiency. If you code a number using binary then each storage location stores a single bit but if you can find a way to store more than just two states at a particular location then you can store more data in the same space.

Put another way, a binary number has more bits in it than the equivalent decimal number has digits. For example, the decimal number 1234 is 10011010010 in binary. Binary numbers take more space and we trade space for simplicity. Some forms of Flash memory, for example, use three or more voltage levels to store data using non-binary bases.

So far no one has gone back to basics to build a decimal computer but one day, when binary computers have run out of steam, perhaps Babbage will be proved right.

For more on binary numbers and two's complement representation in particular, see Binary - negative numbers

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