## Big Idea

Our decimal number system is a base-ten place value system. This system extends infinitely in two directions towards very large numbers and very small numbers.

## Background points for teaching

An understanding of the number system is central to success in mathematics. This tutorial examines 4 mathematical ideas associated with the base-ten place value system. Each of these ideas is described with examples so that teachers can gain a picture of the mathematics they are teaching.

- The places (or columns) in the number system are based around groupings of ten.
- The base ten place value system has a repeating naming pattern.
- The decimal point is a convention that indicates the units place.
- Zero is necessary as a placeholder. For example, 1.08

## The places (or columns) in the number system are based around groupings of ten. For example, 10 ones = 1 ten, 10 hundreds = 1 thousand

One of the most basic ideas within the number system is that there is a 10-to-1 relationship between the values of any two **adjacent** places (or columns). Figure 1 illustrates the x10 relationship between adjacent places as you move to the left.

From Figure 1 you can see that ten of any one place makes one of the next larger place, for example, 10 hundreds = 1 thousand, 10 hundredths = 1 tenth. This ten makes 1 concept continues indefinitely to larger and larger numbers with the relationship between adjacent places remaining the same regardless of which two places are being considered.

The 10-to-1 relationship also allows you to understand the relationship between **any** two places in the number system. For example: The thousands are 2 places to the left of the tens. This means there is a 10x10 (10^{2}) relationship between tens and thousands: 100 tens = 1 thousand. The tens are 3 places to the left of the hundredths. This means there is a 10x10x10 (10^{3}) relationship between tens and thousands: 1000 hundredths = 1 ten.

The 10-to-1 relationship can be viewed as a 1-to-10 relationship between adjacent places to the right where the places are getting smaller by a factor of ten (see figure 2). In other words a division of ten or a tenth of any place makes one of the next smaller place, for example, a tenth of a thousand = one hundred (1000÷10=100), a tenth of a tenth = one hundredth (0.1 ÷10 = 0.01).

The 1-to-10 relationship also allows you to understand the relationship between **any** two places in the number system. For example: The tenths are 3 places to the right of the hundreds. This means there is a ÷10÷10÷10 (10^{-3}) relationship between hundreds and tenths: 1/1000 of a hundred = 1 tenth.

## The base ten place value system has a repeating naming pattern

__Whole Numbers__

The base ten place value number system has a repeating naming pattern. The pattern of ones, tens and hundreds names firstly the units, then the thousands, and then the millions etc.

The convention of including a space between each family or group helps us read the number. For example 67 000 = sixty seven thousand, or 502 000 = five hundred and two thousand.

__Decimals__

The grouping pattern also extends to the right of the decimal point.

The decimals in the naming pattern have equivalent decimals which are easier to read. For example 10 thousandths is equivalent to 1 hundredth. So, although the naming pattern continues to the left of the decimal point the convention is to name the decimal by the place value column furthest to the right. For example, 0.7 is read as 7 tenths, while 0.83 is read as 83 hundredths. Often decimals are simply read as a string of digits, with 0.83 read as zero point eight three. Encouraging students to read decimals by using their place value supports them in their development of number sense.

## The decimal point is a convention that indicates the units place

The role of the decimal point is to indicate the units or ones place in a number and it does that by sitting immediately to the right of that place. Consequently the decimal point also works to separate the units (on the left) from parts of the unit (on the right).

As shown below a number can be expressed in different ways depending on the selection of the unit. The decimal point indicates which position is the unit.

6501.4 (in this case the “ones” is assumed)

650.14 tens (in this case the units are tens)

65014 tenths (in this case the units are tenths)

6.5014 thousands (in this case the units are thousands)

## Zero is necessary as a placeholder. For example, 1.08

Like any other digit, zero indicates the number of items in the place (or column) in which it appears. For example, 205 means 2 hundreds, 0 tens and 5 ones. Without the 0, the 205 appears as 25, which means 2 tens and 5 ones. We refer to the zero as a place holder. In the example it holds the tens place, so that the 2 is correctly located in the hundreds place. Zero is also needed as a place holder in decimal numbers. In both whole numbers and decimals the zero holds the place between a digit and the decimal point. For example in 300 the two zeros hold the ones and tens places so that the 3 is correctly placed in the hundreds place, while in 6.05 the zero holds the tenths place so that the 5 appears in the hundredths place. The zero is not needed as a place holder when it is not between a digit and the decimal point, for example 005 and 5 are the same, as are 1.50 and 1.5.