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

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
____________

Is this correct or not?

```

```
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

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

If this answer didn't contain enough detail for you, or if

- 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?

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
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.

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

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