The Decimal System

The decimal system has ten figures: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

The decimal system, which is also known as base ten, represents every number in powers of ten (units, tens, hundreds etc...).

For example,

345 means 3 hundreds, 4 tens and 5 units
2307 means 2 thousand, 3 hundreds, 0 tens and 7 units

In column notation:

Powers of ten:

Where does the ‘base’ part come from in base ten?

This terminology comes from exponential (and the exponentials inverse, the logarithmic) function.

An example of an exponential is 3⁴. This means 3 x 3 x 3 x 3. That is 3 to the power of 4.

The raised number, 4 here, is called an index or an exponent, and the number itself, 3 here, is called the base.

In general, a^p = a x a x ... to p factors, with a being the base and p the index.

The Binary System

The Binary system of calculation has only 2 figures: 0 and 1.
This is why a solar system maybe referred to as a binary system if it contains two stars.

As with base ten, base two enables us to represent every number in powers of 2 (units, twos, fours, eights, sixteens, etc...).

Powers of two:

2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, ...

The binary system operates in exactly the same way as our decimal system, with the use of powers of two instead of powers of ten.
For example:

111 means 1 four, 1 two and 1 one = 4 + 2 + 1 = 7
1010 means 1 eight, 0 fours, 1 two and 0 ones = 8 + 2 = 10

Just as we use the column headings H T U (hundreds, tens and units) in the decimal system, we can use S E F T U (sixteens, eights, fours, twos and units) for numbers in the binary system. Thus, the above examples can be represented as follows:

When using bases other than base ten, it is necessary to indicate the base using a subscript, e.g.

21_ten = 10101_two
58_ten = 111010_two

To convert a number from base ten to base two, the following method can be used:
Convert 26_ten to base two:

Fortunately, most scientific calculators have the ability to convert numbers between different bases.

The puzzle ‘Guess my number’, which is often included in Christmas crackers, uses base two. With a primary class it can give the teacher the appearance of being a mindreader - something which we are generally expected to have!

Guess my number

Ask a pupil to choose a number between one and fifty and to keep it secret.
Next, ask them to tell you in which sets below the number can be found.

The number that they originally chose is the sum of the numbers found in the top left hand corner of each identified set.

For example:

If a pupil choses the number 26, then they will tell you that their number appears in sets B, C and D. The numbers in the top left corner of these sets are 8, 2 and 16 respectiveley, giving a total of 8 + 2 + 16 = 26! You’ve read their mind!

How does ‘Guess my number’ work?

The sets only contain the numbers 1 - 50. All these numbers can be represented in base two using only the first 6 columns (thirty-twos, sixteens, eights, fours, twos and ones).

Note that a column representation of a number is unique to that number.

Note also there are six sets. Each set contains only numbers which have an entry in a particular column. e.g. In set B, only numbers which use the eights column in base two have an entry.

In this way, when a pupil tells you which sets there number is in, they are really telling you the columns which are used in representing their number.
e.g.
If a pupil tells you that there number lies in sets A, C, E and F, then they are telling you that their number has an entry in the units column, the twos column, the fours column and the thirty twos column.

So by a simple addition, you can read their minds! In the above example, the answer is 39.

This obviously works for numbers upto 63 (which is the largest number that can be represented using only six columns in base two:

63_ten = 111111_two, 64_ten = 1000000_two

To allow pupils to choose the numbers 1 - 63 however, the numbers 51 - 63 will each need to be included in the appropriate sets.

This need not be the end of the game. By including a seventh set whose entries all have an entry in the sixtyfours column, or an eighth set whose entries all have an entry in the one hundred and twenty eights column. This process can be continued ad infinitum, as long as we keep including new sets and including the ever larger new numbers in existing sets.

Beyond base two and ten

This is of course, not the end of the story.

Apart from
The decimal system, also known as the denary system, (base ten) where we count in ones, tens, hundreds, etc... (powers of ten) and
The binary system (base two), where we count in ones, twos, fours, etc... (powers of 2)
there are also
The ternary system (base three), where we count in ones, threes, nines, etc... (powers of 3)
The octal system (base eight) where we count in ones, eights, sixty-fours, etc... (powers of 8)
The duodecimal system (base twelve) where we count in ones, twelves, one hundred and forty fours, etc... (powers of 12). The Latin for twelve is duodecim.
The hexadecimal system (base sixteen) where we count in ones, sixteens, two hundred and fifty sixes etc... (powers of sixteen).

The duodecimal systems everyday use is apparent with time and imperial measurement. We count hours in twelves or twenty fours; half dozen and dozen eggs; 1 gross = 12² = 144. The hexadecimal system has application in computer programming as it is closely connected with the binary sytem.

There is of course a counting system for any number, however the above ones have more common usage.

There is of course a problem . Base ten uses the digits 0 to 9, which are all the digits we have. In displaying numbers in column format, only one digit is permissible in a column. In using systems of a base greater than ten in the hexadecimal system for example, there is no way of writing the number 14_ten. We therefore need to invent new digits. The extra digits in the hexadecimal system are commonly denoted as:

11_ten=A_sixteen, 12_ten=B_sixteen, 13_ten=C_sixteen, 14_ten=D_sixteen, 15_ten=E_sixteen

The hexadecimal system has 16 figures: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E.
We can now write;
20_ten = 14_sixteen = 1 sixteen + 4 units
28_ten = 1B_sixteen = 1 sixteen + 12 units
78_ten = 4D_sixteen = 4 sixteens + 14 units
3119_ten = B2E_sixteen = 12 256’s + 2 sixteens + 15 units

Confused? Again most scientific calculators do convert to base sixteen.

The advantage of the duodecimal system or the hexadecimal system over the denary sysstem is that they require fewer digits to represent a given number, as can be seen in the last example.

The obvious disadvantage of these systems, is that more symbols are necessary than in the denary system.