Living Large with Base Ten

What do you notice or wonder? 

All the symbols above represent the number of tiles, or units, in the collection below.

The symbolic numeral “184” uses the base ten notational structure to represent quantity. Base ten is one of the most commonly used number systems throughout the world.

Grouping 10

Believed to have been developed in India around the 1st century AD, the base ten notational system uses groups of 10 to represent quantity.

Let’s take a closer look at the numeral 184 to see how this works. Each tile below has a value of 1.   

We formed 18 groups of 10 tiles each. Four single tiles remain—not enough to form another group of 10. What is the value of 18 groups of tens? We can say that the numeral 184 represents 18 tens, plus 4 ones. 

Grouping Groups of 10

While we can’t form another group of 10 tiles, we can group 10 tens to form a larger group. How many tiles will be in this larger group?

Ten groups of tens become a square-shaped group containing 100 tiles—a hundreds group. With no other possibility for making more groups of ten, we have arrived at the minimal collection of groups in the base ten system for 184 tiles: 1 hundred, 8 tens, and 4 ones.

• More about minimal collections

A collection of tile or number groups with no more than 9 of any one-sized groups is said to be the minimal collection.  

The 4 pictures below all represent the same value or collection of tiles. Which shows the minimal collection?

In Collections 1, 2, and 3, there are more than 9 of one type of groupings. Therefore, they are not minimal collections. In Collection 4, there are fewer than 10 ones, fewer than 10 tens, and fewer than 10 hundreds. While all the collections accurately show the value of 184 tiles, only Collection 4 is a minimal collection. A minimal collection directly translates to a base ten numeral used to represent the collection symbolically, in this case, 184.

Try creating a minimal and nonminimal collection for 213 in the Number Pieces app.

Number Pieces App

Place Value Notation

The values of digits in base ten numerals are determined by the size of the number groups or tiles the digits represent. The smallest group, the ones, are positioned farthest to the right. To the left of the ones, are the tens groups. To the left of the tens are the hundreds, and so on. The numeral 184 has a value of 1 hundred (100), 8 tens (80) and 4 ones (4).

This place value method of representing quantities with numerals is based on the idea of positional notation. The value of each digit depends on its position in the numeral. 

• Don't all number systems use positional notation?

Actually, many early number systems did not use place value or positional notation, even if they were based on groupings of 10. For instance, Roman numerals assigned different values to different symbols.  Find the Roman numeral at the base of the pyramid on the back of a US one-dollar bill.

Each of these symbols has a base 10 value either added or subtracted based on its position in the sequence:  M = 1000, D = 500, C = 100, L = 50, X = 10, V = 5, and I = 1.

Here, MDCCLXXVI = 1000 + 500 + 100 + 100 + 50 + 10 + 10 + 5 + 1

The year 1776, in which the United States declared independence, is much easier to represent using just 4 digit positions that convey the base 10 value.

What happens if we add 18 ones to our collection of 184? How will that change the value of our number? What will the numeral be for the new minimal collection?

After we finish our grouping process to obtain the minimal collection, we have 2 hundreds and 2 ones. Because there are no tens, it’s tempting—especially for children just learning the place value system—to only record the groupings present.

Previously, we explored the numeral 0 as representing an empty set. In this case, we must use 0 as a placeholder to represent the empty set of tens and record the number as 202.

Representing Larger Numbers

The same grouping-by-ten process can be used for representing much larger numbers as well. If the number has more than 9 hundreds, a thousands group (1,000) could be formed.

If the number had more than 9 thousands, a ten-thousands group (10,000) could be formed.

Any number can be represented in base ten by recording the number and size of each successive grouping of 10 in the minimal collection, from largest to smallest, in sequence, left to right.  

While the visual grouping process can be a bit tedious with large numbers, such experiences can help build a concrete sense of large numbers.  What does it mean if Earth’s population has grown from 3 to 7 billion people within the last 100 years? How does 1 million dollars compare to 1 billion? 

Let’s visualize some large numbers by looking at pennies on a soccer field.

Now that we have a better sense of 1 million pennies, how many pennies do you think it will take to cover the soccer field?

Hopefully this exploration raised additional mathematical wonderings for you. How much is that field of pennies worth in dollars? How many soccer fields would it take to hold a million pennies? A billion pennies?

We have completed the first leg of our journey. Our next leg will explore the world of addition and subtraction. It’s a trip you’ve taken before, but on this journey, we hope you’ll discover new insights and learn new ways of working with numbers.