Welcome to Omni's expanded form calculator - your short article of an option for learning how to compose numbers in increased form. In essence, the expanded type in math (also called expanded notation) is a means to decompose a value into summands equivalent to that digits. The subject is similar to scientific notation, though here, we split it right into even an ext terms. To do the link even clearer, we have three different options of composing numbers in expanded type in the calculator, such as the expanded type with exponents.

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Expanded type is vital in miscellaneous parts of math, e.g., in partial products algorithm. Therefore what is increased form? Well, let's run straight right into the article and also find out!


What is the broadened form?

The expanded kind definition is the following:

💡 writing numbers in expanded kind means mirroring the value of every digit. To be precise, us express the number as a amount of terms the correspond to the digit of ones, tens, hundreds, etc., and also those of tenths, hundredths, and also so on for the expanded type with decimals.

As pointed out above, the expanded notation of, say, 154 need to be a amount of terms, each linked to among the digits. Obviously, we can't just write 1 + 5 + 4 because that's miles away from what us had. So how do you compose a number in expanded form? Well, you include zeros.

154 = 100 + 50 + 4

So what walk expanded kind mean? Intuitively, we associate every digit of the number with something that has actually the very same digit, followed by sufficiently numerous zeros to end up in the ideal position when we amount it all up. To make it much more precise, let's have it neatly explained in a separate section.


How to write numbers in broadened form

Let's take a number that has actually the kind aₙ...a₄a₃a₂a₁a₀, i.e., the aₖ-s denote consecutive digits the the number through a₀ gift the people digit, a₁ the tens digit, and also so on. Follow to the broadened form meaning from the vault section, we'd favor 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 compose such numbers in expanded form starting native the best side, i.e., indigenous a₀. Due to the fact that it is the ones' digit, that must show up at the finish of our number. We create b₀ by writing as countless zeros to the best of a₀ as we have actually digits after a₀ in our number. In various other words, we add none and also get b₀ = a₀.

Next, we have the 10s digit a₁. Again, we form b₁ by placing as many zeros come the right of a₁ as we have actually digits following a₁ in the original number. In this case, there's one together (namely, a₀), therefore we have b₁ = a₁0 (remember that here we usage the notation of writing digit after digit). Similarly, to b₂ we'll include two zeros (since a₂ has a₁ and a₀ come the right), definition that b₂ = a₂00, and so on until bₙ = aₙ00...000 with n-1 zeros.

Alright, we've seen just how to create numbers in expanded form in a special situation - when they're integers. But what if we have actually decimals? Or if it's some long-expression with number of numbers before and after the dot? What is the expanded type of together a monstrosity?

Well, let's see, shall we?


How to write decimals in broadened form

Essentially, we carry out the same as in the over section. In short, us again add a suitable variety of zeros to a digit however for those ~ the decimal dot, we compose them to the left instead of come the right. Obviously, the dot need to be placed at the right spot so the it all makes sense (we can't have an integer beginning with zeros, ~ all). So exactly how do you compose a number in expanded kind when it has some fountain part?

The structure from the very first section doesn't change: the expanded type with decimals need to still provide 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 cₖ-s are digits, when bₖ-s and also dₖ-s room numbers). Fortunately, we achieve bₖ-s likewise as before; we just need to remember come take the dot into account. To be precise, we include as many zeros together we have digits come the right, but prior to the decimal dot (i.e., we just count the a-s).

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

For instance, to find d₁, we take c₁ and add as many zeros together we have between the decimal period and c₁ (which is, in this case, none). Then, we add the symbols 0. at the really beginning, which gives d₁ = 0.c₁. Similarly, we placed one zero come the left of c₂ (since we have actually one digit between the decimal dot and also c₂, namely c₁), and also 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 actually an expanded kind example through the number 154.102:

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

(Note exactly how we have nothing equivalent to the hundredths digit. The is due to the fact that it's equal to 0, so in the increased notation, it would certainly be 0.00, or merely 0, i.e., nothing.)

A keen eye may have actually noticed a typical thread when writing numbers in expanded kind (even the expanded type with decimals): it's all around adding zeros in the right places. What is more, zeros naturally correspond come 10, 100, 1000, and 0.1, 0.01, 0.001, and so on. An also keener eye could observe the 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 at the expanded kind in math: with exponents.


Expanded kind with exponents

Exponents the 10 room very simple. Whenever we take some integer power of 10 (we're no considering portion exponents here), the result is the number 1 v several zeros that corresponds to that power. As we've seen at the end of the above section, the first three hopeful powers are:

10¹ = 10, 10² = 100, 10³ = 1000,

so the outcomes are the number 1 through one, two, and three zeros, respectively. Top top the various other hand, the first three an adverse powers are:

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

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

Another nice residential or commercial property of strength of 10 is that once we multiply any type of of them by a one-digit number, the result is the same thing, but with the 1 replaced by that number. Because that instance:

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

and these look just like the summands we observed in the increased notation. In other words, we can exchange every summand as soon as writing numbers in expanded type with a multiplication of miscellaneous that consists of the digit 1 and also some zeros through a one-digit number. And that defines how to create numbers v decimals in expanded form with factors (note how we can pick such an option in the expanded form calculator).

So what go expanded type mean in this case? it again tells united state to decompose our numbers into summands matching to the digits, however this time, the summands space of the type "digit times a number with 1 and some zeros."

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

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

Using the discussion above, we can equivalently write this expanded type example as:

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

However, we have the right to go also further! Remember exactly how we stated at the beginning of this ar that all these components are powers of 10? Well, let's write them as such! This way, we acquire yet an additional expanded notation: the expanded type with exponents (observe how we can select this choice in the expanded type calculator).

So what is the expanded type with exponents? as before, it's decomposing our number into summands matching to the digits, however now the summands take it the type "digit time 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 strength that appear here agree with the subscripts we used in the 2nd section. Also, note exactly how 1 is additionally a strength of 10, i.e., the zeroth. In fact, any number elevated to power 0 equals 1.

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All in all, we've controlled to learn just how to write numbers v decimals in expanded kind in three different ways: with numbers, through factors, and with exponents.

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


Using the expanded kind calculator

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

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

Easy, isn't it? Also, note how for convenience, the expanded kind calculator list consecutive summands heat by row and doesn't cite the terms the correspond come the digits 0 (similarly to exactly how we did as soon as we learned 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 usage it. It's a an excellent starting allude for learning an ext about numbers and how we stand for them, therefore be sure to examine out Omni's arithmetic calculators section for more amazing tools.