Welcome to Omni's expanded type calculator - your write-up of selection for learning just how to compose numbers in expanded form. In essence, the expanded type in mathematics (also referred to as expanded notation) is a way to decompose a value right into summands equivalent to that is digits. The object is comparable to scientific notation, though here, we separation it right into even more terms. To do the connection even clearer, we have three various options of writing numbers in expanded form in the calculator, such together the expanded form with exponents.

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

## What is the expanded form?

The expanded type definition is the following:

 💡 writing numbers in expanded form means mirroring the worth of each digit. To it is in precise, us express the number together a sum of terms the correspond come the number of ones, tens, hundreds, etc., and those the tenths, hundredths, and so on for the expanded form with decimals.

As pointed out above, the expanded notation of, say, 154 have to be a amount of terms, each connected to among the digits. Obviously, we can't simply write 1 + 5 + 4 since that's miles away from what us 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, us associate every digit the the number v something that has the same digit, followed by sufficiently plenty of zeros to finish up in the right position as soon as we amount it every up. To do it much more precise, let's have it neatly defined in a separate section.

## How to create numbers in expanded form

Let's take a number that has actually the type aₙ...a₄a₃a₂a₁a₀, i.e., the aₖ-s denote consecutive digits that the number through a₀ gift the ones digit, a₁ the 10s digit, and so on. According to the broadened form an interpretation from the vault section, we'd choose to write:

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

with the number (not digit!) bₖ corresponding in which method to aₖ.

Let's describe how to create such number in expanded form starting indigenous the appropriate side, i.e., indigenous a₀. Since it is the ones' digit, the must appear at the end of our number. We create b₀ by creating as plenty of zeros come the best of a₀ as we have digits after a₀ in our number. In other words, we add none and also get b₀ = a₀.

Next, we have the tens digit a₁. Again, we form b₁ by placing as many zeros to the best of a₁ as we have actually digits following a₁ in the initial number. In this case, there's one such (namely, a₀), for this reason we have actually b₁ = a₁0 (remember that here we use the notation of writing digit after ~ digit). Similarly, to b₂ we'll include two zeros (since a₂ has actually a₁ and also a₀ to the right), an interpretation that b₂ = a₂00, and also so on till bₙ = aₙ00...000 v n-1 zeros.

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

Well, let's see, shall we?

## How to compose decimals in expanded form

Essentially, we execute the same together in the above section. In short, we again add a suitable variety of zeros come a digit yet for those after ~ the decimal dot, we compose them to the left rather of to the right. Obviously, the dot should be put 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 frame from the an initial section doesn't change: the expanded form with decimals need to still offer us a amount of the form:

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

(remembing the aₖ-s and also cₖ-s are digits, while bₖ-s and dₖ-s are numbers). Fortunately, we obtain bₖ-s an in similar way as before; us just need to remember come take the dot right into account. To it is in precise, we add as plenty of zeros together we have digits to the right, but prior to the decimal dot (i.e., we just count the a-s).

On the various other hand, we uncover dₖ-s by placing as plenty of zeros on the left side the cₖ-s together we have actually digits between the decimal dot and the digit in question.

For instance, to discover d₁, we take c₁ and include as numerous zeros together we have in between the decimal period and c₁ (which is, in this case, none). Then, we include the symbols 0. at the very beginning, which provides d₁ = 0.c₁. Similarly, we put one zero come the left the c₂ (since we have one digit in between the decimal dot and also c₂, specific c₁), and obtain d₂ = 0.0c₂. Us repeat this for all d-s until dₘ = 0.000...cₘ, which has actually m-1 zeros ~ the decimal dot.

Let's have an expanded kind example v the number 154.102:

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

(Note exactly how we have nothing corresponding to the hundredths digit. The is because it's equal to 0, for this reason in the broadened notation, it would certainly be 0.00, or merely 0, i.e., nothing.)

A to crawl eye may have noticed a typical thread when writing number in expanded form (even the expanded type with decimals): it's all about adding zeros in the appropriate places. What is more, zeros naturally correspond come 10, 100, 1000, and also 0.1, 0.01, 0.001, and so on. An also keener eye might observe that all these numbers room 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 means of looking in ~ the expanded form in math: with exponents.

## Expanded kind with exponents

Exponents of 10 room very simple. Whenever we take part integer strength of 10 (we're no considering fraction exponents here), the an outcome is the digit 1 through several zeros that coincides to the power. Together we've viewed at the finish of the above section, the first three positive powers are:

10¹ = 10, 10² = 100, 10³ = 1000,

so the results are the digit 1 v one, two, and three zeros, respectively. Top top the various other hand, the very first three an adverse powers are:

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

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

Another nice building of strength of 10 is that as soon as we multiply any type of of castle by a one-digit number, the an outcome is the same thing, however with the 1 replaced by the number. For instance:

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

and this look just like the summands we saw in the broadened notation. In various other words, we might exchange every summand when writing numbers in expanded kind with a multiplication of other that consists of the number 1 and some zeros by a one-digit number. And that explains how to write numbers through decimals in expanded type with factors (note exactly how we can select such an alternative in the expanded form calculator).

So what does expanded type mean in this case? it again tells us to decompose our numbers into summands equivalent to the digits, however this time, the summands space of the kind "digit times a number v 1 and also some zeros."

Let's have an example to check out it clearly. Recall indigenous the above section that:

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

Using the debate above, we can equivalently write this expanded kind example as:

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

However, we deserve to go even further! Remember just how we claimed at the start of this ar that every these components are strength of 10? Well, let's compose them as such! This way, we acquire yet an additional expanded notation: the expanded kind with exponents (observe exactly how we can pick this choice in the expanded kind calculator).

So what is the expanded form with exponents? as before, it's decomposing our number right into summands matching to the digits, but now the summands take it the form "digit times 10 to part power." In this new variant, the over expanded form example looks prefer this:

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

Observe exactly how the powers that show up here agree with the subscripts us used in the 2nd section. Also, note exactly how 1 is likewise a strength of 10, i.e., the zeroth. In fact, any number elevated to power 0 equals 1. All in all, we've controlled to learn just how to create numbers through decimals in expanded type in three various ways: v numbers, through factors, and with exponents.

In fact, there's only one thing continuing to be to do: let's complete with describing just how to usage the expanded form calculator.

## Using the expanded type calculator

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

Input the number you'd like to have actually in increased notation into the "Number" field.Choose the form you'd like to have: numbers, factors, or exponents by selecting the right word in "Show the prize in ... Form."Enjoy the result offered to friend underneath.

Easy, isn't it? Also, note just how for convenience, the expanded form calculator lists consecutive summands heat by row and doesn't mention the terms the correspond come the digits 0 (similarly to exactly how we did as soon as we learned just how to create decimals in expanded form).

And that's that.

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We've learned the expanded form definition and how to use it. It's a great starting allude for learning more about numbers and how we stand for them, for this reason be sure to inspect out Omni's arithmetic calculators ar for more awesome tools.