We use our fingers to count like a scale with 10 different levels (technically 11 if we start at zero but early civilisations didn't have a concept of zero).

This is why we use the Base 10 or Denary number system. However, if we use all 10 of our fingers, we are actually representing 1 lot of 10 and 0 units, so we don't really need to use our last finger ...

When we write numbers, we should always include the base they are in.  For instance, the number 1101 is ambiguous - I could assume that it's written in denary, but this may not be the case ...

• 1010₂
• 1010₈
• 1010₁₀
• 1010₁₆

... all represent different number of 'things' (10, 520, 1010 and 4112 things respectively).

Counting on your fingers (See this website)

Normally, when we count, we use each finger to represent just one 'thing'. This can be severely limiting when it comes to counting a large number of things because we quickly run out of fingers! How would it be different if we could use one finger to represent each available symbol in our base? That means we could use one finger to represent each place (and it's associated value) in the number rather than just one value.

Consider, then two situations ...

... which goes some way to explain why computers count in binary.

Activity 2 Why binary? (15)

Task 2.1 Why Binary Engagement in activity After you have carried out the activity, write about what you have found out. OUTCOME : Written explanation of what you have found out from 'Why Binary?' activity

Activity 3 Binary - Denary Conversions (60)

Task 3.1 Binary to Denary conversions Notebooks Brain Convert the following binary numbers to denary. Show your working. 000011012 000101112 000110102 010111002 101101012 OUTCOME : Binary to denary conversions with workings

Task 3.2 Denary to Binary conversions Notebooks Brain Convert the following denary numbers into binary. Show all your working. 2910 3710 9610 7210 1210 OUTCOME : Denary to binary conversions with workings.

Activity 4 Hexadecimal (60)

Task 4.1 Binary to hexadecimal conversion Notebooks Brain Convert (represent) the following binary strings to hexadecimal. Show all your working. 100001102 100100112 1000010001002 0100010010002 101001102 OUTCOME : Binary to hexadecimal conversions with workings.

Task 4.2 Hexadecimal to binary conversion Notebooks Brain Convert the following hexadecimal numbers to binary. Show all your working. 8316 A916 57216 60416 ABC16 OUTCOME : Hexadecimal to binary conversions with working.

How about these?

1. Binary to denary conversion program
   
   Write a program to input a binary number as a text string, convert it to denary and output the result. You may use the following algorithm:
   
   Output "Enter a binary number "
Input BinaryString 
BinaryLength ← Length of BinaryString
Multiplier ← 1
DenaryValue ← 0
For n ← BinaryLength Down To 1
    Digit ← nth character in BinaryString
    DenaryValue ← DenaryValue + Digit * Multiplier
    Multiplier ← Multiplier * 2
Next n
Output "The denary equivalent is ", DenaryValue
   
   Note: Line seven requires a function that returns the nth character from a string. It may also need a function to convert this character to an integer.
   
   Could you adapt this so that it can handle different number bases?
   
   
2. Denary to binary conversion program
   
   Write a program to input a denary number, convert it to binary and output the result. You may use the following algorithm which employs integer division:
   
   Output "Enter a denary number "
Input DenaryValue
BinaryString ← ""
While DenaryValue > 0
    BinaryString ← CStr(DenaryValue MOD 2) + BinaryString
    DenaryValue ← DenaryValue DIV 2
Endwhile
Output "The binary equivalent is ", BinaryString
   
   Note: The function CStr converts a number to a string. The sign ‘+’ here represents concatenation.
   
   
3. Poster competition
   
   Create a poster about binary : research and include simple binary rules, jokes, binary watches / clocks, comics …
   
   
4. The Octal number system
   
   One other commonly used number system is Octal, or Base 8, since octal can be used to represent 3 bits where hexadecimal is used to represent 4. Read more about Octal on it's Wikipedia page. One particularly interesting fact is that the Yuki people from America count in Octal because they count using the gaps between the fingers rather than the fingers themselves.