Place Value and Decimal Calculations

What place value means

Our number system is based on ten. A digit does not have a fixed value by itself: its value depends on its position. In $4000$, the digit 4 represents four thousands, but in $0.04$ it represents four hundredths.

Starting at the units column, each move one column to the left makes a digit worth ten times as much:

$1,\ 10,\ 100,\ 1000,\ 10000,\ldots$

Each move one column to the right makes it worth one tenth as much:

$1,\ 0.1,\ 0.01,\ 0.001,\ldots$

The same pattern therefore connects very large and very small numbers.

Reading and partitioning large numbers

Commas can separate a large whole number into groups of three digits. Reading from the right, these groups are units, thousands, millions and then billions. For example,

$3,472,905=3,000,000+400,000+70,000+2,000+900+5.$

This is called partitioning the number. The zero in the tens column matters: it shows that there are no tens and keeps the 9 and 5 in their correct positions.

To write “three million, forty thousand and six” in figures, preserve every empty column with a zero:

$3,040,006.$

Omitting any of the necessary zeros changes the place values and does not represent the stated number.

Place value after the decimal point

The decimal point is a separator, not a place-value column. It separates whole units from parts of a unit: the units and tenths positions are adjacent, with the point between them. To its right are tenths, then hundredths, thousandths and ten-thousandths. There is no “oneths” column.

In $6.375$, the digits represent

$6+\frac{3}{10}+\frac{7}{100}+\frac{5}{1000}.$

The 7 therefore has a value of $0.07$, not $0.7$. Similarly, the 8 in $0.0508$ is worth $0.0008$ because it is in the ten-thousandths column.

Zeros at the end of a decimal do not alter its value, so

$0.5=0.50=0.500.$

These zeros represent no extra hundredths or thousandths. By contrast, the zero in $0.05$ is essential: it places the 5 in the hundredths column, making $0.05$ one tenth of $0.5$.

Comparing decimal values

Compare decimals from left to right, beginning with the largest place-value column. It can help to add trailing placeholder zeros so that every number has the same number of decimal places.

For example, write $0.6$, $0.45$, $0.099$ and $0.5$ as

$0.600,\quad 0.450,\quad 0.099,\quad 0.500.$

The tenths digits show that $0.099$ is smallest. Comparing the remaining values column by column gives

$0.099<0.45<0.5<0.6.$

The number of visible decimal digits is not a reliable guide to size: $0.099$ has more digits than $0.45$, but it is smaller.

Multiplying and dividing by powers of 10

Multiplying by $10$ moves every digit one place-value column to the left; multiplying by $100$ or $1000$ moves every digit two or three columns left. Dividing has the opposite effect.

For instance,

$3.6\times100=360$

because the 3 moves from units to hundreds and the 6 moves from tenths to tens. A placeholder zero is needed in the units column. Also,

$452\div1000=0.452,$

because each digit moves three columns to the right.

It is better to describe the digits as changing columns rather than saying that the decimal point moves. The decimal point remains fixed between the adjacent units and tenths columns. A digit crossing that boundary has moved one place, not two. This view also works when zeros must be inserted, as in

$0.006\times1000=6.$

Aligning decimals in addition and subtraction

Digits with the same place value must be combined. Therefore, line up decimal points before adding or subtracting, and use placeholder zeros where useful:

  12.50
+  0.37
-------
  12.87

Here, hundredths are added to hundredths, tenths to tenths and units to units. Lining up the final digits instead would mix different place values.

The same principle applies to subtraction:

  8.200
- 0.465
-------
  7.735

Writing $8.2$ as $8.200$ does not change its value, but it makes the thousandths, hundredths and tenths columns clear.

Place value in decimal multiplication and division

Place value explains where the decimal belongs in a product. Since $2.4$ is 24 tenths and $0.3$ is 3 tenths,

$2.4\times0.3=24\times3\text{ hundredths}=72\text{ hundredths}=0.72.$

For division by a decimal, multiplying both numbers by the same power of 10 can make the divisor a whole number without changing the quotient:

$4.8\div0.6=48\div6=8.$

Both numbers were multiplied by 10, so their relative size stayed the same.

Finally, use magnitude as a check. Multiplying a positive number by $0.3$ should make it smaller, so an answer of $7.2$ for $2.4\times0.3$ cannot be sensible. Place-value checks help detect answers that are out by a factor of $10$, $100$ or $1000$.