Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Base 2, Base 8 Multiplication and Addition


Date: 10/28/2001 at 19:53:33
From: Barbara 
Subject: Binary multiplication and adding

Hello, Dr. Math!

When you add binary numbers and you have 4 one's, so four in binary is 
0100, and you add 1+1+1+1 = 4 but in binary is 0 and carry 0 or carry 
1, for example this is multiplication and addition:

        11011
       x10111
    __________
        11011
       11011       
      11011
     00000
    11011
  +__________
   111001101
 
Next is also multiplication and addition, for example:

                    10.01
                   x 1.01111
                  ____________
My answer is:     101.1000111 

Is this correct or not?

Thank you in advance for your help and explanation.


Date: 10/29/2001 at 04:02:28
From: Doctor Jeremiah
Subject: Re: Binary multiplication and adding

Hi Barbara,

The problem is that you can't add up more than two columns without 
considering what would happen if you had a carry greater than 1. This 
actually happens in your first question. You did the multiplication 
right, but when you added the five rows of numbers it should have gone 
something like this:

Up to here it's fine:

 carry ->    1
 carry ->    1111
              11011
             11011
            11011
           00000
        + 11011
        -----------
              01101

Notice there are two carries because

    1+1 = 0 + 1 carry
 +  1+1 = 0 + 1 carry
---------------------
     4  = 0 + 2 carries

If we use both carries it continues like this:

 carry ->   11
 carry -> 1111111
              11011
             11011
            11011
           00000
       +  11011
        -----------
         1001101101

Which is the right answer, because

         11011        27
       x 10111      x 23
    ----------      ----
    1001101101       621

In your second question we need to to it the same way as with decimal: 
first count the decimals in the answer, then multiply them as if there 
were no decimals, then put in the decimal point afterward:

               1001
           x 101111
        -----------
               1001
              1001
             1001
            1001
           0000
          1001

Now we get to the addition:

carry ->   1111
               1001
              1001
             1001
            1001
           0000
          1001
        -----------
          110100111

Which makes our answer:

         10.01
        x 1.01111
        -----------
         11.0100111

This makes sense because: 10.01 is just slightly greater than 2, 
1.01111 is just slightly greater than 1, and the answer 11.0100111 is 
just slightly greater than 3

Imagine in decimal:  2.3 x 1.4 = 3.22

Your answer was larger than 5, and the only way to make that is for 
either both numbers to be greater than 2, or for one of them to be 
greater than 3. It always pays to check your answers.

If this answer didn't contain enough detail for you, or if
you just want to talk about this some more, please write back.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/   


Date: 10/29/2001 at 17:13:28
From: Barbara Shaker
Subject: Re: Binary multiplication and adding

Thank you very much for your answer. It's enough and was very nicely 
explained. Now I understand that I can't have more than two carries. 
number.  

I have another question for you: when I do multiplication in octal I 
multiply in decimal and then convert to octal, but I wonder about
the carry. For example:
    
           EF12
           3456 (base 16)
        __________
          59A6C 
         4AB5A
        3BC48
     + 2CD36
    ____________
       30DFF80C  

                           4407
                          x2356  (base 8)
                         ________
                          33052
                         30443
                        15425
                     + 15016
                     _____________  
                       17120202    

Is that correct or not? Can I do the multiplication this way? 

Thank you in advance for your answer and your help.
Barbara


Date: 10/29/2001 at 19:20:55
From: Doctor Jeremiah
Subject: Re: Binary multiplication and adding

Hi Barbara,

All bases are done the same way. You do the mutliplication and then 
the addition with however many carries you need (the carries cannot 
exceed the base the question is in.) The same problem can happen in 
decimal. Add up 12 nineteens. The answer for the first column is 8 
with a carry of 10, but you can't have a carry greater than 9, so you 
would have to have two carries: 9 and 1

  carry ->  1
  carry -> 29
            19
            19
            19
            19
            19
            19
            19
            19
            19
            19
            19
         +  19
          ----
           228

Anyway, the question in hex was fine. I didn't see any problems. In 
the octal one you accidentally got the wrong answer for your 
multiplication. Here is the answer I got:

                           4407
                         x 2356  (base 8)
                       --------
                          33052
                         26443
                        15425
                       11016

And when you do the addition you only have to worry if your carry 
exceeds 7. It doesn't, so you get:

                        12211
                          33052
                         26443
                        15425
                     + 11016
                     ----------
                       13100202

Here is how I miltiply things in bases other than 10. It works for me 
but it may not seem like a good way to you:

In order to multiply bases other than 10 and 2, what you need to do is 
something in relation to base 10.

For example, in octal 7*6 is the same as 7*5+7, and since octal 7 is 
just before the rollover (like decimal 9), in decimal the equivalent 
would be 9*5+7 = 52. That is the same answer as octal 7*5+7.

In octal, 4*7 is the same as 4*6+4, and since octal 4 is halfway to 
the rollover (like decimal 5), in decimal the equivalent would be 
5*6+4 = 34. That is the same answer as octal 4*6+4.

In the end, you need to check your work. If you have access to a 
computer with Microsoft Windows you can click the start button and 
choose programs and then accessories and start the calculator. Make 
sure the calculator is in "scientific mode" on the view menu. Now you 
can choose any base you want out of 2, 8, 10, and 16. If you do a 
calculation or type a number in with one base, selecting a different 
base will convert the answer.

Let me know if you want more explanation or if you have other 
questions.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/