>Sets and Systems>What Is the Place-value System?

What Is the Place-value System?

A number in the decimal numeral system is made up of digits, placed in different combinations. The structure of the numbers follows a system. That system is what we call the place-value system.

The place-value system is built up in such a way that each place where you can put a digit has a certain value. You can use as many places as you want to, like the thousandths place and the millionths place. Here are the different decimal places:

Thousandths place: Place-value $0.001$
Hundredths place: Place-value $0.01$
Tenths place: Place-value $0.1$
Ones place: Place-value $1$
Tens place: Place-value $10$
Hundreds place: Place-value $100$
Thousands place: Place-value $1000$
Ten thousands place: Place-value $10 000$

A number where the place value for each digit is indicated.

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Example 1

In the number $64$ , the $6$ is in the tens place and the $4$ is in the ones place. You can split up the number by using the values from the place-value system. That means that you have the place-value the same number of times as the digit in the place. It becomes like this:

64 = 6 tens and 4 ones

Meaning that you have

$64$
$↙$ $↘$
$60 and 4$

Example 2

In the number $100$ , $1$ is in the hundreds place, $0$ in the tens place and $0$ in the ones place. You can split up the number by using the values from the place-value system. That means you have the place-value the same amount of times as the digit in the place. It becomes like this:

\begin{array}{l} 100 & = 1 hundreds and 0 tens \\ and 0 ones \end{array}

100 = 1 hundreds and 0 tens and 0 ones

Meaning that you have

$100$
$↙$ $↓$ $↘$
$100 and 0 and 0$

Example 3

In the number $0.42$ , $0$ is in the ones place, $4$ is in the tenths place and $2$ in the hundredths place. You can split up the number by using the values from the place-value system. That means you have the place-value the same amount of times as the digit in the place. It becomes like this:

\begin{array}{l} 0.42 & = 0 ones + 4 tenths \\ + 2 hundredths \end{array}

0.42 = 0 ones + 4 tenths + 2 hundredths

Meaning that you have

0.42 = 0 + 0.4 + 0.02

The Structure of the Place-value System

All digits to the left of the ones place are multiplied by $10$ one or more times.

All digits to the right of the point are divided by $10$ one or more times.

A number moves to the next place in the place-value system when it’s multiplied by $10$ The names of the places in each direction continue like this infinitely.

It’s very useful to understand how the place-value system is structured. As you grow older, you’ll come across extremely large and small numbers that go way past what you saw above.

Example 4

This number has no decimal point:

2341

But, there’s an invisible point behind the last digit. When you don’t have any decimals, you don’t have to write the point. The last digit in the number is in the ones place.

Example 5

Look at the number $62.7$

Here the $6$ is in the tens place, $2$ is in the ones place and $7$ is in the tenths place. You can write the number $62.7$ as a sum to show the reasoning:

\begin{array}{l} 62.7 & = 60 + 2 + 0.7 \\ = 6 \times 10 + 2 \times 1 + 7 \times 0.1 \end{array}

Example 6

Look at the number $5349.728$

Here $5$ is in the thousands place, $3$ is in the hundreds place, $4$ is in the tens place, $9$ is in the ones place, $7$ is in the tenths place. $2$ is in the hundredths place and $8$ is in the thousandths place. You can write the number $5349.728$ as a sum to show the reasoning:

\begin{array}{l} 5349.728 & = 5000 + 300 + 40 + 9 \\ + 0.7 + 0.02 + 0.008 \\ = 5 \times 1000 + 3 \times 100 + 4 \times 10 \\ + 9 \times 1 + 7 \times 0.1 \\ + 2 \times 0.01 + 8 \times 0.001 \end{array}

\begin{array}{l} 5349.728 & = 5000 + 300 + 40 + 9 + 0.7 + 0.02 + 0.008 \\ = 5 \times 1000 + 3 \times 100 + 4 \times 10 + 9 \times 1 \\ + 7 \times 0.1 + 2 \times 0.01 + 8 \times 0.001 \end{array}

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