Conversions

Decimal

We count in what is called the decimal counting system. Decimal has 10 digits which are 0,1,2,3,4,5,6,7,8,9. Decimal is also called base 10 because it has 10 digits. The reason why people started counting in decimal is because it has 10 digits and we have 10 fingers and people used to use their fingers for counting.

Binary

Binary uses only 2 digits which are 0 and 1 and is also called base 2. Binary is what computers use for counting because inside a computer you get things that are like little switches that can be either off or on(0 or 1). A subscripted 2 is placed after a binary number so that it can be recognised as being a binary number like in the following example:

1001₂

We already know that when we count we add 1 to the current number each time to get the next number. If we start at 0 and then add 1 we get 1. Now we have to add 1 to 1 but we have run out of digits because there are only 2 digits in binary. When we get to 9 in decimal and add 1 we first change the number to a 0 and then add 1 to the imaginary 0 to the left of it which gives us 10. If we apply this to binary then we change the 1 to 0 and then increase the imaginary 0 to the left of it which gives us 10 but this 10 is actually 2. The following sequence shows you how to count to 5 in binary:

0 1 10 11 100 101

Hexadecimal

The word hexadecimal is made up of 2 parts which are hex(6) and decimal(10). If you add 6 and 10 together you get 16 and that is how many digits there are in hexadecimal. Hexadecimal is sometimes called hex or base 16. To get 16 digits we have to use letters of the alphabet and those 16 digits are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. Hexadecimal is often used instead of binary numbers because just 2 hexadecimal digits can make the same numbers as 8 binary digits which in turn make up a byte. A subscripted 16 is used to show that a number is hexadecimal:

7F₁₆

Octal

Octal uses 8 digits which are 0,1,2,3,4,5,6,7 and is also called base 8. We use a subscripted 8 to show that a number is octal:

275₈

Converting from decimal to binary, hexadecimal and octal

decimal -> binary₂

5 -> binary₂

First write a few 2s(because it's base 2) to the power of the numbers 0,1,2,3,...

2³ 2² 2¹ 2⁰

Next calculate the values of these numbers.

8 4 2 1

The highest number must be greater than the number you want to convert.

Now divide the number to be converted by the number starting on the left. Write how many times the number divides into it underneath the number and then write your remainder on top of the next number to be divided. You then divide by the remainder each time after that.

5 5 1 1
8 4 2 1
0 1 0 1

The bottom line of numbers is the binary number. Make sure to leave out all leading 0s

5 -> 101₂

decimal -> hexadecimal₁₆

Decimal to hexadecimal works in the same way except that you use 16 to the power of those numbers since that is the base this time.

26 -> hexadecimal₁₆

16² 16¹ 16⁰

26 26 10
256 16 1
0 1 10

Before we write the answer we must change all numbers greater than 9 to their hexadecimal equivalents.

1 10
1 A

26 -> 1A₁₆

decimal -> octal₈

This time we use 8 to the power of those numbers because we are working with base 8.

12 -> octal₈

8² 8¹ 8⁰

12 12 4
64 8 1
0 1 4

12 -> 14₈

Converting from binary, hexadecimal and octal to decimal

binary₂ -> decimal

1101₂ -> decimal

First write 2 to the power of the numbers 0,1,2,3,...

2³ 2² 2¹ 2⁰

Now change them to their real values.

8 4 2 1

Now put the binary number underneath the other numbers and multiply the top number by the number beneath it and put the answer underneath the other 2 with + between each number and add the bottom row together to get your final answer.

8 4 2 1
1 1 0 1
8+4+0+1 = 13

1101₂ -> 13

hexadecimal₁₆ -> decimal

Once again you use 16 instead of 2.

1F₁₆ -> decimal

16¹ 16⁰

16 1
1 F
16+F = 16 + 15 = 31

1F₁₆ -> 31

octal₈ -> decimal

25₈ -> decimal

8¹ 8⁰

8 1
2 5
16+5 = 21

15₈ -> 21

Converting from hexadecimal to binary

hexadecimal₁₆ -> binary₂

2C₁₆ -> binary₂

First write the number as decimals.

2 12

Now divide each number by the numbers 8 4 2 1.

2 12
8421 8421
0010 1100

Take the leading 0s away and you have your answer.

2C₁₆ -> 101100₂

binary₂ -> hexadecimal₁₆

10111₂ -> hexadecimal₁₆

First separate the binary number into groups of 4 digits. If the number at the front has less than 4 digits then add 0s to the front of it.

0001 0111

Now use the numbers 8421 and multiply each time and add the results together for each group to get the answer.

0001 0111
8421 8421
0+0+0+1 = 1; 0+4+2+1 = 7

10111₂ -> 17₁₆

Converting from octal to binary

Octal to binary uses the same method as hexadecimal to binary except that you use the numbers 421 and groups of 3 digits.

octal₈ -> binary₂

26₈ -> binary₂

2 6
421 421
010 110

26₈ -> 10110₂

binary₂ -> octal₈

1010₂ -> octal₈

001 010
421 421
0+0+1 = 1; 0+2+0 = 2

1010₂ -> 12₈

Converting from hexadecimal to octal

We do not convert directly from hexadecimal to octal but instead first convert to binary and then to octal.

hexadecimal₁₆ -> octal₈

45₁₆ -> octal₈

First convert the hexadecimal number to binary.

4 5
8421 8421
0100 0101

Now make groups of 3 digits and then convert to octal.

001 000 101
421 421 421
0+0+1 = 1; 0+0+0 = 0; 4+0+1 = 5

45₁₆ -> 105₈

octal₈ -> hexadecimal₁₆

27₈ -> hexadecimal₁₆

First convert the octal number to binary.

2 7
421 421
010 111

Make groups of 4 and then convert to hexadecimal.

0001 0111
8421 8421
0+0+0+1 = 1; 0+4+2+1 = 7

27₈ -> 17₁₆