Back

Our common denary system

Before we look at binary, we need to understand how the numbering system we use day-to-day works. This numbering system that we commonly use every day of our lives is known as the denary system (although people often call it the 'decimal system' as well). It's also known as the 'base 10' system. This is because there are ten digits in use. These are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. If we have just one digit then, for example, we know that the symbol 7 is ‘worth more’ than the symbol 3 because that is how we have defined these symbols to be. However, it is not only the size of the number that tells us something about the number. It is also the position of each digit relative to any other digits in the same number.

For example, in the number 268, the 6 is worth a lot more than the 8 because it is to the left of the 8. Similarly, the 2 is worth a lot more than both the 8 and the 6 because it is on the very left of the number. Can you remember when you first learnt to count? You probably would have used headings to start with, to help you understand that the position of each digit is important to the 'worth' of that digit. Any number e.g. 3892 would have been written down under the headings, as shown here.

This means the denary number is ‘worth’ (3 x 1000) + (8 x 100) + (9 x 10) + (2 x 1) which adds up to 3892. 

Q1. Show how the number 7283 is made up using the table below.

Q2. Show how the denary number 149 is made up using the table below.

Q3. How many different symbols are there in the denary system? What are they?
Q4. After units, then tens, then hundreds, then thousands, what is the next number?
Q5. There is actually a pattern for calculating the 'worth' of each position in a denary number. Calculate the following:

10⁰ = 
10¹ = 
10² = 
10³ = 
10⁴ = 
10⁵ = 
10⁶ = 
10⁷ = 

and so on. When we count in denary, we start at 0 and then go 1, 2, 3 etc up to 9. We don't have a symbol for ten, so we 'carry one' and then return the units to 0. We then carry on counting, like this ...

1
2
3
4
5
6
7
8
9
10
11
12
13....

19
20
21
22
23 ....

98
99
100
101

Q6. Some students always seem to be confused by the result of powers of 0 and powers of 1. Use a calculator to work out the following:

10⁰
5⁰
45⁰
6⁰
2⁰
1⁰

Q7. What is any number to the power of 0?

Q8. Use a calculator to work out the following:

201
131
893431
21
23431
11

Q9. What is any number to the power of 1?
Q10. Where does the word 'denary' originally come from?
Q11. Where does the word 'binary' originally come from?
Q12. If denary uses a total of 10 different digits, how many digits does the binary system use? What are they?

Back