56 lines
1.8 KiB
Markdown
56 lines
1.8 KiB
Markdown
---
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tags:
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- Theory_of_Computation
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- electronics
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- binary
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---
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Now that we know how to add and multiply using binary numbers we can apply this knowledge to our previous understanding of circuits.
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Our aim will to be have our inputs as the numbers that we will add or multiply on and our outputs as the product or sum.
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## Half adder
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Let's start with the most basic example:
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*Half adder circuit*
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This circuit has the following possible range of outputs, where A and B are the input switches and X and Y are the output signals. The logic gates (an `XOR` and an `AND` ) are equivalent to the add function.
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````
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A B X Y
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_ _ _ _
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0 0 0 0
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0 1 0 1
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1 0 0 1
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1 1 1 1
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````
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We can see that if we treat A and B as single binary digits that could correspond to $2^1$ (either `0` or `1` ) , then the X and Y outputs can be viewed collectively to constitute the sum of A and B (we have put the denary equivalent in brackets)
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````
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A B X Y
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_ _ _ _
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0 0 0 0 0 + 0 = 00 [0]
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0 1 0 1 0 + 1 = 01 [1]
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1 0 0 1 1 + 0 = 01 [0]
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1 1 1 1 1 + 1 = 10 [2]
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````
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This is called a half adder because it cannot go higher than $2^1$.
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### Representing binary output as denary values
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There are special output components that can represent the combination of binary inputs and logic gates as denary values. Here is an example using a **seven-segment display** :
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[maths_with_logic_gates_5.gif.crdownload](../img/maths_with_logic_gates_5.gif.crdownload)
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## Full adder
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To represent numbers higher than the denary 2, we would need a carrying function so that we could represent numbers up to denary 3 and 4. The limit of a half adder is $2^1$.
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We do this by adding another switch input:
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