# Place Value Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Steven Wooding
Last updated: Jan 18, 2024

Table of contents:

Calculating the place value is something we do instinctively, at least for small numbers. But what is exactly the math behind this operation? Are there rules? Discover with Omni the interesting world of the most fundamental math: you may end up intrigued! Keep reading to learn:

• What is the place value chart?
• How to calculate the decimal place chart?
• Calculating the place value chart with decimals (it's easier than you think!)
• How to use and read our decimal place value calculator.

## What is the positional notation?

The positional notation is a fundamental concept of arithmetic, one on which we lay our entire understanding of math. The core idea of the positional notation is that, regardless of the base with which you are writing a number, every digit is independent and associated with one and only one factor.

The factors are powers of the base. By arranging them in ascending order, we create a chart of multipliers. By multiplying a digit by the factor in the corresponding position and summing all the results, we find the true value of a number.

## How to calculate the place value chart in the decimal number system

It's harder to explain how to calculate the place value in words than it is using an example. Let's walk through it: we'll do it with the decimal number system.

In the decimal number system, we deal with base $10$. This means there are ten possible digits. You know them: $0$, $1$, $2$, ...

The last digit has a value of $9$. What's the following number? $10$: there are two digits! One of them ($0$) is in the same position as the previous ones and has a value of $0$. The other one is one position to the left and has a value of $1$. How do we associate a value to the number $10$? Check this formula:

$\footnotesize 10 = 1\cdot 10 + 0$

Or, if we want to be pedantic:

$\footnotesize 10 = 1\cdot 10^1 + 0\cdot 10^0$

Each individual digit has its own multiplier, which is nothing but the base raised to the exponent equal to the position of the digit in the number.

This notation is easily scalable for larger numbers:

$\footnotesize \begin{split} 18,\!452&=1\cdot10^4+8\cdot10^3+4\cdot10^2\\ &\quad +5\cdot10^1+2\cdot10^0\\[1em] &=1\cdot10,\!000+8\cdot1000\\ &\quad+4\cdot100+5\cdot10 +2 \end{split}$

🙋 If you see a link to the scientific notation, you are not wrong! Visit the scientific notation calculator to learn more about this!

The place value chart notation allows you to spell numbers in a rather easy and direct way (at least for some languages). You need to assign a name to every factor and then patch together the name of the digit and this name. Take the first three factors:

• $10^0=1$ reads as ones;
• $10^1=10$ reads as tens; and
• $10^2=100$ reads as hundreds.

Now take a number like $457$, where we meet the digits four, five, and seven. Join the two lists:

• $7\cdot1$Seven ones.
• $5\cdot10$Five tens.
• $4\cdot 100$Four hundreds.

There are specific words for tens in English, and we can omit the ones in the word: the number becomes four hundred fifty-seven. Did you see the hyphen? It appears for every number between $21$ and $99$.

For numbers larger or equal to $1000$, we need to add a different noun for every new group of three factors.

• For the factors between 4 and 6, we add the noun thousand.
• Between factors 7 and 9, we use million.
• Next step, billions for factors 10 to 12.

We can find as many groups as we need, but they get increasingly unmanageable. It's worth mentioning trillions and quadrillions, though.

Knowing all these rules, we can build compound nouns from the first three (ones, tens, and hundreds) and the next group. The first nine names we use in the decimal place value chart calculations are:

Factor

Name

1

Ones

10

Tens

100

Hundreds

1000

Thousands

10,000

Ten thousands

100,000

Hundred thousands

1,000,000

Millions

10,000,000

Ten millions

100,000,000

Hundred millions

🙋 The place value chart is of fundamental importance when you need to convert from any base to the decimal base. Try our binary converter and see if you can compete with a computer at multiplications and sums!

## After the period: find the place value chart with decimals

Calculating the place value chart for decimal numbers is super easy, barely an inconvenience! The only differences lie in the direction with when we read the digits and the names we assign to the factors.

• The direction is from left to right, moving away from the decimal points (mirror direction than the one we used for the integer part).
• The names reflect the fact that we are dealing with fractions of the base. In the following table, you can meet them.

Factor

Name

0.1

Tenths

0.01

Hundredths

0.001

Thousandths

0.0001

Ten-thousandths

0.00001

Hundred-thousandths

0.000001

Millionths

To spell a decimal number, spell the decimal part as you'd do for an integer number and add the name of the factor where it ends. Separate the integer part with an "and". For example, we'd read the number $0.21459$ as zero and twenty-one thousand four hundred fifty-nine hundred thousandths.

🙋 Things may get a bit hairy in the case of periodic decimals... Do you know them? Meet them at the interesting terminating decimals calculator 😉

## How to use our decimal place value calculator

Our decimal place value calculator is a straightforward tool. Input the number of your interest, and let us calculate the place value chart. At the top of the calculator, you'll see a table containing the digits and the related factors. Below the table, you can see how we'd spell the number.

🙋 We are adding languages to this tool! Go to advanced mode and explore how numbers are spelled in other languages around the world!

And if you want to play more with digits, try our digit sum calculator. It does something pretty similar to this tool but with an interesting twist!

## FAQ

### How do I find the place value chart in base 10?

To find the place value chart in base 10, follow these easy steps for an integer number:

1. Split the number into single digits.
2. Reverse the digits, or read them from right to left.
3. Assign to every digit a power of 10 with the exponent equal to the position. The first position has a value of 0.
• The first digit will be the ones;
• The second will be tens,
• The third will be the thousands.
4. Proceed until you run out of digits.

That's it!

### What is the place value chart of 1568.23?

The place value chart of 1568.23 is:

• 1 thousands;
• 5 hundreds;
• 6 tens;
• 8 ones; and
• 2 tenths;
• 3 hundredths.

Notice how reading the table gives an approximation of the proper spelling of the number: one thousand five hundred sixty-eight and twenty-three hundredths.

### What do I calculate from the place value chart in other bases?

From the place value chart in a given numerical base, you can find the decimal representation of any number. To do so, multiply every digit of a number in a given base for the power of the base with the exponent equal to the digit's position in the number starting from the right to the left, with initial position 0. By summing the results, you will find the representation in base 10 of the number.

### What is a place value chart?

A place value chart is a way to represent a number by dividing it into single digits, each associated with a specific factor related to the numerical base in which we are working.

The factors for decimal numbers are: ones, tens, hundreds, thousands, ten thousands, etc. Each of them multiplies the digit in the corresponding position: for example, in the number 6358, we find:

• Ones multiplying 8;
• Tens multiplying 5;
• Hundreds multiplying 3; and
• Thousands multiplying 6.
Davide Borchia
Number
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