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And Bit Multiplication

author
James Smith
• Monday, 02 November, 2020
• 7 min read

Before going through this section, make sure you understand about the representation of numbers in binary. You can read the page on numeric representation to review.

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Contents

This document will introduce you to the methods for adding and multiplying binary numbers. In each section, the topic is developed by first considering the binary representation of unsigned numbers (which are the easiest to understand), followed by signed numbers and finishing with fractions (the hardest to understand).

Adding unsigned numbers in binary is quite easy. Addition is done exactly like adding decimal numbers, except that you have only two digits (0 and 1).

So to add the numbers 06 10 =0110 2 and 07 10 =0111 2 (answer=13 10 =1101 2) we can write out the calculation (the results of any carry is shown along the top row, in italics). In this case the extra carry from the most significant bit has no meaning.

With signed numbers there are two ways to get an overflow -- if the result is greater than 7, or less than -8. Obviously both of these results are incorrect, but in this case overflow is harder to detect.

But you can see that if two numbers with the same sign (either positive or negative) are added and the result has the opposite sign, an overflow has occurred. Typically, DSP's, including the 320C5x, can deal somewhat with this problem by using something called saturation arithmetic, in which results that result in overflow are replaced by either the most positive number (in this case 7) if the overflow is in the positive direction, or by the most negative number (-8) for overflows in the negative direction.

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There is no further difficult in adding two signed fractions, only the interpretation of the results differs. This is very useful because it means we can use the same circuitry for addition, regardless of the interpretation of the results.

Even the generation of overflows resulting in error conditions remains unchanged (again compare with above) Multiplying unsigned numbers in binary is quite easy.

Multiplication can be performed done exactly as with decimal numbers, except that you have only two digits (0 and 1). When multiplying larger numbers, the result will be 8 bits, with the leftmost set to 1, as shown.

As long as there are n+m bits for the result, there is no chance of overflow. The easiest is to simply find the magnitude of the two multiplicands, multiply these together, and then use the original sign bits to determine the sign of the result.

Multiplication by zero is a special case (the result is always zero, with no sign bit). Although it is almost impossible to get an overflow (since the multiplicands and results usually have magnitude less than one), it is possible to get an overflow by multiplying -1x-1 since the result of this is +1, which cannot be represented by fixed point numbers.

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This means that we end up with two bits to the left of the decimal point. This obviously presents a difficulty if we wanted to store the number in a Q3 result, because if we took just the 4 leftmost bits, we would end up with two sign bits.

So what we'd like to do is shift the number to the left by one and then take the 4 leftmost bit. The other three fundamental operations are addition, subtraction and division.

In the case of a binary operation, we deal with only two digits, i.e. 0 and 1. The operation performed while finding the binary product is similar to the conventional multiplication method.

The four major steps in binary digit multiplication are: And no additional number is borrowed or carried forward in this operation.

In this section, you will get answers for the questions about binary multiplication, including: This process has to be continued until all the multiplier is done, and finally, the addition operation is made.

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The multiplication by 1 makes all the multiplicand value unchanged. As per these rules, it is very clears, that if the binary multiplication includes 0, then it will result in zero itself.

To differentiate the rules for multiplication of binary numbers from other binary operations such as addition, subtraction and division, please refer to below given table: Some binary multiplication examples are given below for a better understanding of this concept.

Another example of binary multiplication with a decimal point is as follows: Solve the below-given questions using multiplication rules, based for binary numbers.

For more information on binary operations like addition, subtraction, and division operations register with BYJU’S -The Learning App and also watch exciting videos to learn with ease. Binary multiplication, unlike decimal subtraction, involves only two digits, i.e. 0 and 1.

Visit BYJU’S to learn everything about binary multiplication. To solve multiplication problems based on binary numbers, we have to use the four primary rules designated for this operation.

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Just like the adder and the subtractor, a multiplier is an arithmetic combination logic circuit. In the end, we add the digits while keeping in mind the carry that might generate.

Based on the above equation, we can see that we need four AND gates and two to design the combination circuit for the multiplier. The AND gates will perform the multiplication, and the half adders will add the partial product terms.

If you would like to brush up your knowledge of digital logic gates, we’ve got you covered FAM. Multiplying the two numbers with each other using standard binary arithmetic rules, we get the following equation.

Note: We will use all the equations above when we code multipliers in our BHDL and Verizon course. He is currently pursuing a PG-Diploma from the Center for Development of Advanced Computing, India.

Related Hex Calculator | IP Subnet Calculator The binary system is a numerical system that functions virtually identically to the decimal number system that people are likely more familiar with. Apart from these differences, operations such as addition, subtraction, multiplication, and division are all computed following the same rules as the decimal system.

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Almost all modern technology and computers use the binary system due to its ease of implementation in digital circuitry using logic gates. It is much simpler to design hardware that only needs to detect two states, on and off (or true/false, present/absent, etc.).

DecimalBinary0011210311410071118100010101016100002010100While working with binary may initially seem confusing, understanding that each binary place value represents 2 n, just as each decimal place represents 10 n, should help clarify. Reading from right to left, the first 0 represents 2 0, the second 2 1, the third 2 2, and the fourth 2 3 ; just like the decimal system, except with a base of 2 rather than 10.

Find the largest power of 2 that lies within the given number Subtract that value from the given number Find the largest power of 2 within the remainder found in step 2 Repeats until there is no remainder Enter a 1 for each binary place value that was found, and a 0 for the rest 2 n 2 4 2 3 2 2 2 1 2 0 Instances within 1810010Target: 1818 – 16 = 22 – 2 = 0Converting from the binary to the decimal system is simpler.

The only real difference between binary and decimal addition is that the value 2 in the binary system is the equivalent of 10 in the decimal system. Note that the superscripted 1's represent digits that are carried over.

A common mistake to watch out for when conducting binary addition is in the case where 1 + 1 = 0 also has a 1 carried over from the previous column to its right. Note that the superscripts displayed are the changes that occur to each bit when borrowing.

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Binary multiplication is arguably simpler than its decimal counterpart. Note that in each subsequent row, placeholder 0's need to be added, and the value shifted to the left, just like in decimal multiplication.

The complexity in binary multiplication arises from tedious binary addition dependent on how many bits are in each term. Typically, the 0 placeholder is not visually present in decimal multiplication.

While the same can be done in this example (with the 0 placeholder being assumed rather than explicit), it is included in this example because the 0 is relevant for any binary addition / subtraction calculator, like the one provided on this page. Without the 0 being shown, it would be possible to make the mistake of excluding the 0 when adding the binary values displayed above.

Note again that in the binary system, any 0 to the right of a 1 is relevant, while any 0 to the left of the last 1 in the value is not. The dividend is still divided by the divisor in the same manner, with the only significant difference being the use of binary rather than decimal subtraction.

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Sources
1 www.cse.wustl.edu - https://www.cse.wustl.edu/~garnett/cse515t/fall_2019/
2 www.cse.wustl.edu - https://www.cse.wustl.edu/~garnett/cse515t/spring_2017/
3 ocw.mit.edu - https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-90-computational-methods-in-aerospace-engineering-spring-2014/probabilistic-methods-and-optimization/monte-carlo-methods/1690r-monte-carlo-analysis/