What is a Base Ten System?

Having spent so much time thinking about, teaching and researching place value, I am often using the term 'Base 10'. In this blog I wanted to unpack what we mean when we refer to a 'Base 10' place value system. 

We use a Base 10 place value system to represent quantities. This base was likely chosen as we have 10 fingers and toes. Other bases are still commonly used in our society. For example, when recording time, we use a Base 60 system: 60 seconds is one minute, 60 minutes is one hour. Similarly, in the ICT space, a Base 2 binary system is used. In essence, a base indicates the grouping size used to organise and represent a collection.

The Base 10 system means we organise collections of items into groups of ten. We are then able to record the number of groups of ten in the ‘tens column’. For example, if we were counting a collection of 56, we would group in tens and record the number of ‘groups of ten’ in the ‘tens’ column, in this case, five. The remaining ones would be recorded in the ones column, in this case six. The numeral that represents this collection would be written as 56. This translates as 5 groups of tens and 6 ones.

In the Base 10 system we use 10 unique symbols, known as digits to represent every number from the extremely large to the infinitesimally small. The digits we use in Base 10 are: 0,1,2,3,4,5,6,7,8,9. These digits have different values depending on the column in which they are situated.

If we want to represent a collection of 123 items using the Base 10 system, we once again group in tens. We would find 12 tens and 3 ones. However, the conventions of the Base 10 system state that we cannot record more than 9 units in any one column. Instead, we must group again, this time we group the ‘groups of ten’. In this example, we would gather the 10 tens, forming 1 hundred, and record this quantity in the hundreds column. We would record the 2 remaining tens in the tens column and the 3 ones in the ones column.  This would be represented in the numeral 123.

It is important to note that the value of a digit in the hundreds column is 10 times larger than those in the tens column. This is because a digit in the hundreds column is actually representing a count of ‘10 groups of ten’. This pattern continues throughout the number system, with each column increasing in value by a power of ten from right to left. This is known as the Base 10 property of our place value system.

An understanding of the Base 10 property is important as, amongst other things, it provides us with a simple way to multiply and divide by powers of ten. For example, if we multiply 45 by 10, we simply need to move the digit 4 to the hundreds column (where it automatically becomes ten times larger) and the digit 5 to the tens column (where it automatically becomes ten times larger). We then place a zero in the ones column to show there are no ones. This simple act is much more efficient than completing an algorithm and is a perfect application of the Base 10 property.

Our Base 10 place value system is simple but elegant. Beneath this simplicity lies a complex multiplicative structure that is difficult, but critical, for students to understand.

Have a great week!

Ange🎲🎓