can we multiply large numbers using binary system?

Question

I want to multiply 23 and 43 using binary sytem.Is it easier than ordinary multiplication?

Answer 1

Its not easier bu it is possible. Multiplication consists of adding and shifting digits. For your 43 x 23

43 = 101011
23 = 10111

Now to multiply, first I will add some zeros to the front of 43 just to make the book keepping easier:

43 = 0000101011

Now look at the digits in 23. Multiplying any numbe by 23 in binary is:

x * 23 = x *1 + x * 10 + x * 100 + x * 10000

I've just separated out eacjh individual bit in the bianry of 23. The key to multiplying in binary is that it is done by shifting only. For example 1101 x 1000 = 1101000. You just add the zeros to the numbe, just like multiplying by 1000 in decimal. So your multiplication is just adding the shifted numbers. 43 x 23 is then adding up the following:

0000101011
0001010110
0010101100
1010110000

Of course adding in binary can be tough too. Carrying the numbers happens much more often in binary. The right column adds to 1, that's simple. The next one adds to 2 which in binary is 10 so you write a zero and carry the 1. The next column adds to 3 which is 11 in binary (including the previously carried 1) so you write a 1 and carry a 1, etc. etc. Below is the addition:
__________
0000101011
0001010110
0010101100
1010110000
__________
1111011101

Converting back to deciaml to check:

1 + 0*2 + 1*4 + 1*8 + 1*16 + 0*32 + 1*64 + 1*128 + 1*256 + 1*512= 989

Its a job to do, but, if you remember, learning to multiply in decimal was no picnic either.

Answer 2

There is a system, often called something like russian peasant multiplication, that is effectively binary multiplication, and works like this:

Write the larger number, and then the smaller number. Then, make two columns below these numbers by multiplying the larger number by two and dividing the smaller by two each time you make another row. Round down when necessary. Then, add all the numbers in the left column that are next to an odd number in the right column, and stop when you reach 1 in the right column.

Observe for this problem
4323
8611
1725
3442
6881

23, 11, 5, and 1 are odd, so find 43 + 86 + 172 + 688.
The answer is 989, which is the value of 43*23.
It is harder to do the way I just showed, obviously.

In binary,
101011_2
*10111_2
0000101011
0001010110
0010101100
1010110000
----------------
1111011101_2 = 1+4+8+16+64+128+256+512=989
which is the same answer. This, too, is hard, but is effectively how computers will normally do it.

Answer 3

For computers it is easy because they are programmed to do math in binary. For people it would not be easier because numbers would be long and hard to deal with. However, you can break the binary numbers into groups of 4 digits and it would be like doing it in base 16 where 0000=0, 0001=1, 0010=2, 0011=3, 0100=4, 0101=5, 0110=6, 0111=7, 1000=8, 1001=9, 1010=A, 1011=B, 1100=C, 1101=D, 1110=E, 1111=F. This is called the Hexidecimal system. If we multiply 23 by 43 in binary we would get 10111 * 101011 or 17*2B so then we get 10*20+10*B+7*20+7*B=200+B0+E0+4D=3DD (this is equal to 3*256+13*16+13 which is indeed 989 or 23*43)or 1111011101(binary). Anyways it's very confusing for someone who is used to base ten because you would have to learn a whole new set of multiplication tables.

Answer 4

How To Multiply Large Numbers

Answer 5

23 in binary is 10111 and 43 is 101011. If you multiply by 101011 there 6 multiplications to carry out for a start which leads to 6 rows that have to be added after which you have to convert back to a base ten number.

Conclusion
For humans, binary is not easier!

Answer 6

Yes and NO.

Yes, if you are very good and proficient ALREADY in the binary sytem. You still have to remember the rules of binary additions and multiplication. ( I am not aware of any school teaching binary in ELHI as regular calculation method). But then, if you have been doing it since kindergarten, then hey, probably ok.
In that case, you can offer your services for profit. Maybe tough, though - a lot of Made in China calculators doing the same binary operations can be had for as cheap as $1.00 or FREE.

No, if it is taking you a lot more time to complete the calculation than using base ten or decimal multiplication.

Answer 7

you can multiply any two no. in binary system, but it will be easier for you to multiply in decimal, because you are using less no. of digits, while if you using machine, then it will be the same for the machine, since it will convert your decimal no. to binary no. by default.

Answer 8

Of course, you can!
in binary system,
23=10111
43=101011
you can multiply binary numbers same as ordinary multiplication?

101011
*) 10111
----------------------------
101011
101011
101011
000000
+) 101011
------------------------------
1111011101
in ordinary number 1111011101=989=23*43

Answer 9

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Answer 10

yes