Understanding n-bit and m-bit Numbers

[Bourbon daddy]

Hi guys,

I sure this is an astonishingly dumb question, but I am new to embedded systems, so don't be too harsh.

I am taking embedded systems in final year at uni and working through some introductory tutorial sheets.

One question asks;

-If two n-bit numbers are added together, what memory size is required to hold the result?

-If two n-bit numbers are multiplied together, what memory size is required to hold the result?

-If an n-bit number is multiplied with an m-bit number, what memory size is required to hold the result?

What is an n-bit number and how does it differ to an m-bit number? I assume they are just to variable integers, and at first I presumed they could just be any size, but I would not know how to express the answer if this were the case.

 

[rcgldr]

m and n just represent some integer value. For a computer, the common values would be 8, 16, 32, 64. What's missing from these questions is if the numbers are signed or unsigned.

 

[chiro]

The best way to understand this is through logs (in base 2).

If you have n bits for a word then you have 2^n possible values.

For addition this means you need to calculate log_2(2^n + 2^m) and round up where necessary. If m = n then you have the situation where log_2(2^m + 2^n) = log_2(2*2^m) = log_2(2^(m+1)) = m+1 bits.

For multiplication you have log_2(2^m * 2^n) = log_2(2^(m+n)) = m + n bits.

You can use the same sort of rules for all other operations to find the number of bits.

 

[Bourbon daddy]

Ok, that makes sense, I have just assigned a number to n and m as it makes it easier for me to follow the logic and it calculates as you would expect.

Thanks for the help guys, much appreciated!

 

[LudusRex]

Hi there,

First of all, there is no such thing as a dumb question when it comes to learning something new. It's great that you are taking an interest in embedded systems and working through tutorials to improve your understanding.

To answer your question, an n-bit number refers to a binary number with n digits. This means that it can have a maximum value of 2^n - 1. For example, a 4-bit number can have a maximum value of 15 (2^4 - 1 = 15). On the other hand, an m-bit number would have m digits and a maximum value of 2^m - 1.

Now, let's look at the questions:

- If two n-bit numbers are added together, the result will also be an n-bit number. This means that the memory size required to hold the result will be the same as the memory size required for the two initial numbers. For example, if you add two 4-bit numbers, the result will also be a 4-bit number.

- If two n-bit numbers are multiplied together, the result will be a 2n-bit number. This means that the memory size required to hold the result will be double the memory size of the initial numbers. For example, if you multiply two 4-bit numbers, the result will be an 8-bit number.

- If an n-bit number is multiplied with an m-bit number, the result will be an (n+m)-bit number. This means that the memory size required to hold the result will be the sum of the memory sizes of the two initial numbers.

I hope this clarifies the concept of n-bit and m-bit numbers for you. Keep up the good work in your studies!