Subtraction using 10’s complement

In digital computer systems, arithmetic operations are simplified using the radix complement system also known as r’s complement system. The r stands for radix which is a base for a number in a particular number system. In this post, you learn to do subtraction using 10’s complement.

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You must be familiar with the complement system in digital logic to understand this subtraction method. To learn about complements visit the following link.

Digital Design – Complements

Examples of number system are decimal, binary, octal, hexadecimal. In a binary system we have complements.

For example,

If you talk about a binary system, the base is 2, then we have two types of r’s complement.

  1. r’s complement
  2. (r-1)’s complement

For decimal number the r’s complement is 10’s complement and (r-1)’s complement is 9’s complement because base is 10. In other words, decimal number has base r = 10, so  10’s complement and r-1 = 9, so 9’s complement. The binary number has base r = 2, 2’s complement and r-1 = 1 , so one’s complement.

Q1. Subtract using 10’s complement 52 – 12 .

We know that 52  – 12 = 40
Let   m = 52   and  n = 12
Take 10’s complement of 12

\begin{aligned}
&+99\\
&-12\\
&----\\
&+87
\end{aligned}

Now, 87 is 9’s complement because we subtracted it with 99. To make it 10’s complement add 1 to 87. The 10’s complement of 12 is 88. Add the 88 to m

\begin{aligned}
&+52\\
&+88\\
&----\\
&\hspace{8px}140
\end{aligned}

Answer: Remove the extra 1 and you get 40

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Check the second example.

Q2: Subtract using 10’s complement 12 – 52

We know that 12 < 52, so answer is -40
Let m = 12 and n = 52
10’s complement of 52

\begin{aligned}
&+99\\
&-52\\
&----\\
&+47
\end{aligned}

The 9’s complement of 52 is 47. To make it 10’s complement add 1 to 47.
Add 48 to m

\begin{aligned}
&+12\\
&+48\\
&----\\
&+60
\end{aligned}

This is not the answer , wait

Take one more 10’s complement of the result.

\begin{aligned}
&+99\\
&-60\\
&----\\
&+39
\end{aligned}

The 9’s complement of 60 is 39. Add 1 to 39 and make is 10’s complement.
Add negative to the result because m > n.
Answer:  -40

Summary

How do I take 10’s complement  ?

Suppose n = 123 then
There are 3 digits in 123. The 10’s complement would be 9’s complement + 1.

\begin{aligned}
&999 - 123 = 876\\
&876 + 1 = 877
\end{aligned}

Therefore, the 10’s complement of 123 is 877.

References

  • John.F.Wakerly. 2008. Digital Design: Principles And Practices, 4/E. Pearson Education, India.
  • Mano, M. Morris. 1984. Digital Design. Pearson.

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