# Expanded Form Calculator

By Maciej Kowalski, PhD candidate
Last updated: Sep 09, 2021

Welcome to Omni's expanded form calculator - your article of choice for learning how to write numbers in expanded form. In essence, the expanded form in math (also called expanded notation) is a way to decompose a value into summands corresponding to its digits. The topic is similar to scientific notation, though here, we split it into even more terms. To make the connection even clearer, we have three different options of writing numbers in expanded form in the calculator, such as the expanded form with exponents.

Expanded form is crucial in various parts of math, e.g., in partial products algorithm. So what is expanded form? Well, let's jump straight into the article and find out!

## What is the expanded form?

The expanded form definition is the following:

 💡 Writing numbers in expanded form means showing the value of each digit. To be precise, we express the number as a sum of terms that correspond to the digit of ones, tens, hundreds, etc., as well as those of tenths, hundredths, and so on for the expanded form with decimals.

As mentioned above, the expanded notation of, say, `154` should be a sum of terms, each connected to one of the digits. Obviously, we can't just write `1 + 5 + 4` since that's miles away from what we had. So how do you write a number in expanded form? Well, you add zeros.

`154 = 100 + 50 + 4`

So what does expanded form mean? Intuitively, we associate each digit of the number with something that has the same digit, followed by sufficiently many zeros to end up in the right position when we sum it all up. To make it more precise, let's have it neatly described in a separate section.

## How to write numbers in expanded form

Let's take a number that has the form `aₙ...a₄a₃a₂a₁a₀`, i.e., the `aₖ`-s denote consecutive digits of the number with `a₀` being the ones digit, `a₁` the tens digit, and so on. According to the expanded form definition from the previous section, we'd like to write:

`aₙ...a₄a₃a₂a₁a₀ = bₙ + ... + b₄ + b₃ + b₂ + b₁ + b₀`,

with the number (not digit!) `bₖ` corresponding somehow to `aₖ`.

Let's explain how to write such numbers in expanded form starting from the right side, i.e., from `a₀`. Since it is the ones' digit, it must appear at the end of our number. We create `b₀` by writing as many zeros to the right of `a₀` as we have digits after `a₀` in our number. In other words, we add none and get `b₀ = a₀`.

Next, we have the tens digit `a₁`. Again, we form `b₁` by putting as many zeros to the right of `a₁` as we have digits following `a₁` in the original number. In this case, there's one such (namely, `a₀`), so we have `b₁ = a₁0` (remember that here we use the notation of writing digit after digit). Similarly, to `b₂` we'll add two zeros (since `a₂` has `a₁` and `a₀` to the right), meaning that `b₂ = a₂00`, and so on until `bₙ = aₙ00...000` with `n-1` zeros.

Alright, we've seen how to write numbers in expanded form in a special case - when they're integers. But what if we have decimals? Or if it's some long-expression with several numbers before and after the dot? What is the expanded form of such a monstrosity?

Well, let's see, shall we?

## How to write decimals in expanded form

Essentially, we do the same as in the above section. In short, we again add a suitable number of zeros to a digit but for those after the decimal dot, we write them to the left instead of to the right. Obviously, the dot must be placed at the right spot so that it all makes sense (we can't have an integer starting with zeros, after all). So how do you write a number in expanded form when it has some fractional part?

The framework from the first section doesn't change: the expanded form with decimals should still give us a sum of the form:

`aₙ...a₄a₃a₂a₁a₀.c₁c₂c₃...cₘ = bₙ + ... + b₄ + b₃ + b₂ + b₁ + b₀ + d₁ + d₂ + d₃ + ... + dₘ`,

(remembing that `aₖ`-s and `cₖ`-s are digits, while `bₖ`-s and `dₖ`-s are numbers). Fortunately, we obtain `bₖ`-s similarly as before; we just have to remember to take the dot into account. To be precise, we add as many zeros as we have digits to the right, but before the decimal dot (i.e., we only count the `a`-s).

On the other hand, we find `dₖ`-s by putting as many zeros on the left side of `cₖ`-s as we have digits between the decimal dot and the digit in question.

For instance, to find `d₁`, we take `c₁` and add as many zeros as we have between the decimal dot and `c₁` (which is, in this case, none). Then, we add the symbols `0.` at the very beginning, which gives `d₁ = 0.c₁`. Similarly, we put one zero to the left of `c₂` (since we have one digit between the decimal dot and `c₂`, namely `c₁`), and obtain `d₂ = 0.0c₂`. We repeat this for all `d`-s until `dₘ = 0.000...cₘ`, which has `m-1` zeros after the decimal dot.

Let's have an expanded form example with the number `154.102`:

`154.102 = 100 + 50 + 4 + 0.1 + 0.002`.

(Note how we have nothing corresponding to the hundredths digit. That is because it's equal to `0`, so in the expanded notation, it would be `0.00`, or simply `0`, i.e., nothing.)

A keen eye may have noticed a common thread when writing numbers in expanded form (even the expanded form with decimals): it's all about adding zeros in the right places. What is more, zeros naturally correspond to `10`, `100`, `1000`, and `0.1`, `0.01`, `0.001`, and so on. An even keener eye might observe that all these numbers are powers of `10`:

`10¹ = 10`, `10² = 100`, `10³ = 1000`, `10⁻¹ = 0.1`, `10⁻² = 0.01`, `10⁻³ = 0.001`.

That brings us to a new way of looking at the expanded form in math: with exponents.

## Expanded form with exponents

Exponents of `10` are very simple. Whenever we take some integer power of `10` (we're not considering fraction exponents here), the result is the digit `1` with several zeros that corresponds to that power. As we've seen at the end of the above section, the first three positive powers are:

`10¹ = 10`, `10² = 100`, `10³ = 1000`,

so the results are the digit `1` with one, two, and three zeros, respectively. On the other hand, the first three negative powers are:

`10⁻¹ = 0.1`, `10⁻² = 0.01`, `10⁻³ = 0.001`,

so again, the digit `1` with one, two, and three zeros, respectively, with the slight change that the zeros appear to the left instead of right (that's a result of the minus in the exponent).

Another nice property of powers of `10` is that when we multiply any of them by a one-digit number, the result is the same thing, but with the `1` replaced by that number. For instance:

`10 * 5 = 50`, `1000 * 3 = 3000`, `0.001 * 6 = 0.006`,

and these look just like the summands we saw in the expanded notation. In other words, we could exchange every summand when writing numbers in expanded form with a multiplication of something that consists of the digit `1` and some zeros by a one-digit number. And that explains how to write numbers with decimals in expanded form with factors (note how we can choose such an option in the expanded form calculator).

So what does expanded form mean in this case? It again tells us to decompose our numbers into summands corresponding to the digits, but this time, the summands are of the form "digit times a number with `1` and some zeros."

Let's have an example to see it clearly. Recall from the above section that:

`154.102 = 100 + 50 + 4 + 0.1 + 0.002`.

Using the argument above, we can equivalently write this expanded form example as:

`154.102 = 1*100 + 5*10 + 4*1 + 1*0.1 + 2*0.001`.

However, we can go even further! Remember how we said at the beginning of this section that all these factors are powers of `10`? Well, let's write them as such! This way, we obtain yet another expanded notation: the expanded form with exponents (observe how we can choose this option in the expanded form calculator).

So what is the expanded form with exponents? As before, it's decomposing our number into summands corresponding to the digits, but now the summands take the form "digit times `10` to some power." In this new variant, the above expanded form example looks like this:

`154.102 = 1*10² + 5*10¹ + 4*10⁰ + 1*10⁻¹ + 2*10⁻³`.

Observe how the powers that appear here agree with the subscripts we used in the second section. Also, note how `1` is also a power of `10`, i.e., the zeroth. In fact, any number raised to power `0` equals `1`. All in all, we've managed to learn how to write numbers with decimals in expanded form in three different ways: with numbers, with factors, and with exponents.

In fact, there's only one thing remaining to do: let's finish with describing how to use the expanded form calculator.

## Using the expanded form calculator

The rules governing the expanded form calculator are straightforward. You just need to follow these three steps:

1. Input the number you'd like to have in expanded notation into the "Number" field.
2. Choose the form you'd like to have: numbers, factors, or exponents by selecting the right word in "Show the answer in ... form."
3. Enjoy the result given to you underneath.

Easy, isn't it? Also, note how for convenience, the expanded form calculator lists consecutive summands row by row and doesn't mention the terms that correspond to the digits `0` (similarly to how we did when we learned how to write decimals in expanded form).

And that's that. We've learned the expanded form definition and how to use it. It's a good starting point for learning more about numbers and how we represent them, so be sure to check out Omni's arithmetic calculators section for more awesome tools.

Maciej Kowalski, PhD candidate
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