Computer Math Part 5: Octal

Octal, or base-8, is commonly used in Unix style operating systems. This number system, being base-8, uses digits 0-7. It’s easily translated to binary, but hex and decimal are a little harder.

Each digit in octal is 3 digits in binary, so 08 is 0002, 18 is 0012, … 78 is 1112. Because it is simple to convert octal to binary, I would suggest that you convert octal to binary and then to decimal or hex.

Unix style operating systems use octal to define file permissions. Each file has a 3 octal digit code. The first digit defines the user’s permissions. The second defines the group’s permissions. The final digit defines everyone else’s permissions.

These permissions are based in binary. The first digit is to allow reading of the file. The second digit is to allow writing to the file. The third digit is to allow execution of the file.

A file with permissions of 777 allows everyone to do everything, while 664 allows the owner and group to read and write, but everyone else only gets to read the file.

Check back Wednesday for a discussion of binary truth tables.

Home work:

Convert the following from octal to decimal:
7778
3778
6648
12345678
76543218
1118

Computer Math, Part 4: Hexadecimal

Now that you understand binary, let’s move onto hexadecimal, or base 16. This is the numbering system most programmers use, because it translates easily to and from binary. Also, 2 hexadecimal digits make up a byte, or 8 bits.

How do we have 16 digits? Simple, hexadecimal uses the digits 0-9 and the letter A-F. 0-9 is the same in hexadecimal as decimal. The digits A-F in hexadecimal are 10-15 in decimal. Here is a conversion chart of single digit hexadecimal:

HexadecimalBinaryDecimal
01602010
11612110
216102210
316112310
4161002410
5161012510
6161102610
7161112710
81610002810
91610012910
A16101021010
B16101121110
C16110021210
D16110121310
E16111021410
F16111121510

Hexadecimal, sometimes simply called hex, is very easy to translate to and from binary. This is because each digit in hex is 4 digits in binary. All you have to do is replace the hexadecimal digit with the binary equivalent. For instance, the number FF16 is 111111112.

Converting a binary number to hex works just slightly different. When you convert a number from hex to binary, you can work from left to right or right to left. When you convert a number from binary to hex, you must work from right to left. This is because a binary number might not be the correct length to work from left to right.

That’s all for today, check back Friday for Octal, or base 8. Here’s some homework:

Convert the following hexadecimal numbers to binary and decimal:
A216
FFFFFF16
10016
CCCCCC16
A2B16
FA6F16
12316
32116
100016
1000016

Computer Math Part 3: Adding Binary numbers

Since it is Monday, I won’t ask you to think too much today. Adding binary numbers is actually easier than adding decimal numbers, if you can believe that.

Here’s how to do it. Write 210 binary numbers down, say 11112 and 1112. Make sure to align them on the right, same as you would to add decimal numbers. This should be on your paper/notepad:

 1111
+ 111

Starting from the right, add the first two digits. 12 + 12 is 102, so write 0 as the right most digit and carry the 1.

Moving on to the next digit, 12 + 12 + 12(carried) = 112, or 310, so write 1 as the next digit and carry the 1.

Third digit is the exact same. 12 + 12 + 12(carried) = 112, or 310, so write 1 as the next digit and carry the 1.

For the Forth digit, 12 + 12(carried) is 102, so write 0 as the right most digit and carry the 1. Drop down the carry, because 1 + 0 is 1 no matter what base you are working with. There’s the answer.

 1111
+ 111
-----
10110

Checking the solution is as simple as opening Windows calculator. If you’re not in scientific view, switch to it from the view menu. Select “Bin” for binary mode. Then put in the math problem, same as any other. Now that you know about calculator, try not using it.

Vocab: A digit in binary can be refered to as a bit, so 64-bit, means 64 binary digits.

Homework:

Create 8 or 9 random binary addition problems. If you want to make it hard, write out 64 digits, or bits, for each binary number.

Computer Math Part 2: Converting Decimal to/from Binary

Wednesday, I discussed the very basics of binary and how to count, or increment, in binary. Today I’ll be discussing converting decimal numbers to and from binary numbers.

Before I cover conversion, let’s talk for a second about bases. That’s what this series boils down to. I’m not talking about military bases, but number bases. Binary is base 2, meaning that there are 2 digits. Because decimal uses 10 digits, it is base 10.

When a number is of a certain base, you denote that by putting a subscripted 2, or 10, immediately after the number. So 1002 means 100 in binary, while 10010 means 100 in decimal. If you don’t fully grasp this, I’m sure you will once you finish the exercises for today.

So, we all know that 12 is 110, but what does 102 equal in base 10? It’s 210. Check out this chart:

BinaryDecimal
12110
102210
1002410
10002810
1000021610
10000023210
100000026410
10000000212810
100000000225610

For every extra zero on the binary side, the decimal side doubles. This is because binary is base 2. You can say that, counting from the right in binary, each digit placement is worth double the previous.

You can use this chart to convert a binary number to a decimal number. For every 1, add the decimal equivilent. For instance, the number 1112 is 110 + 210 + 410 which is 710. Another example is 10012, which is 810 + 110, or 9.

OK, converting binary to decimal is the easier part for today. On to converting decimal to binary. You do pretty much the same thing as converting binary to decimal, but in reverse. The first step is to grab a scrap piece of paper, or open notepad.

To walk through the process, I’ll convert 10510 to binary. The first step is to find the decimal number in the chart that is closest to 10510 without going over, which is 6410. 6410 is 10000002, so write that binary number on the first line. The second step is to subtract 6410 from 10510, which is 4110.

Then we repeat. So, 3210, or 1000002, is the next number. Write 1000002 on the second line, making sure to line up the numbers on the right side, same as you would for decimal. Then subtract 3210 from 4110, which is 910.

Then we repeat. So, 810, or 10002, is the next number. Write 10002 on the second line, making sure to line up the numbers on the right side, same as you would for decimal. Then subtract 810 from 910, which is 110.

Then we repeat. So, 110, or 12, is the next number. Write 12 on the second line, making sure to line up the numbers on the right side, same as you would for decimal. Then subtract 110 from 110, which is 010.
Because, we have reached zero, we can move on the last step. On your paper/notepad should be:

1000000
100000
1000
1

All you have to do is combine these. Starting on the left, any column that has a 1 in it, write a 1, or if the column only has zeros, write a 0. This yeilds: 11010012. Converting this back to decimal will let you know you did it right.

Homework:

Convert the following binary numbers to decimal:
1001
1101
10
10000011
10101010
10110010
100011

Convert the following decimal numbers to binary:
37
123
251
6
192
168
68

Remember to convert it back to check your work. Check back Monday when I’ll discuss adding binary numbers.

Computer Math

Usually when people think of what goes on inside a computer, they think of little ones and zeros zipping around their own tiny raceway at the speed of light. For the most part, that’s correct. Today, I’ll discuss what binary numbers are.

Binary is simply a number system that only has two digits, 0 and 1. The number system that we are taught all throughout school is decimal, which has ten digits, 0 thru 9. There are other number systems that are used when working with computers as well; hexadecimal uses 16 digits, 0-9 and the letters a-f, while octal uses eight digits, 0-7. Any whole number can be easily represented in any of these numbering systems.

Here’s the binary equivalent of the decimal numbers 0-10:

Decimal012345678910
Binary011011100101110111100010011010

You may notice that in binary, 1 + 1 = 10. That’s because you have to carry the 1, just like 1 + 9 = 10 in decimal. Once you can wrap your mind around this, you’ll be on your way to understanding binary. Like everything else in life, it just takes time and practice.

Below are some binary numbers to increase by 1:
1001
1000001001001
11111
111000
10101110
101001

Try to add 1 to each of those numbers, if you find it difficult, then re-read the post and try again. If you find it easy, try to count from 1 to 100000000 in binary. BTW, 100000000 is only 256 in decimal.

Check back in a few days for the next part in the series: conversion from decimal to binary and back to decimal.