Welcome come Omni's expanded kind calculator - your write-up of an option for learning how to compose numbers in broadened form. In essence, the expanded kind in mathematics (also called expanded notation) is a way to decompose a value right into summands corresponding to that digits. The topic is similar to clinical notation, despite here, we separation it 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 together the expanded kind with exponents.
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Expanded kind is crucial in miscellaneous parts that 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 increased form?
The expanded type definition is the following:
|💡 composing numbers in expanded kind means mirroring the value of each digit. To be precise, us express the number together a amount of terms the correspond come the digit of ones, tens, hundreds, etc., as well as those that tenths, hundredths, and also so on for the expanded type with decimals.|
As pointed out above, the expanded notation of, say, 154 must be a amount of terms, each associated to among the digits. Obviously, us can't simply write 1 + 5 + 4 because that's miles far from what us had. So just how do you compose a number in expanded form? Well, you include zeros.
154 = 100 + 50 + 4
So what go expanded type mean? Intuitively, us associate every digit the the number v something that has actually the exact same digit, followed through sufficiently many zeros to finish up in the best position as soon as we sum it all up. To do it much more precise, let's have it neatly defined in a separate section.
How to write numbers in increased form
Let's take it a number that has the kind aₙ...a₄a₃a₂a₁a₀, i.e., the aₖ-s denote continually digits that the number through a₀ gift the people digit, a₁ the 10s digit, and also so on. Follow to the expanded 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 explain how to compose such number in expanded type starting from the best side, i.e., from a₀. Since it is the ones' digit, the must show up at the end of our number. We produce b₀ by writing as numerous zeros to the best 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 10s digit a₁. Again, we type b₁ by putting as plenty of zeros come the right of a₁ as we have actually digits following a₁ in the initial number. In this case, there's one such (namely, a₀), so we have actually b₁ = a₁0 (remember that here we usage the notation of writing digit ~ digit). Similarly, come b₂ we'll add two zeros (since a₂ has a₁ and a₀ to the right), definition that b₂ = a₂00, and also so on until bₙ = aₙ00...000 with n-1 zeros.
Alright, we've seen exactly how to write numbers in expanded form in a special instance - when they're integers. Yet what if we have actually decimals? Or if it's some long-expression with several numbers before and also after the dot? What is the expanded type of such a monstrosity?
Well, let's see, chandelier we?
How to create decimals in expanded form
Essentially, we carry out the same together in the over section. In short, we again add a suitable variety of zeros to a digit yet for those ~ the decimal dot, we compose them come the left rather of come the right. Obviously, the dot must be inserted at the ideal spot so the it all provides sense (we can't have actually an integer starting with zeros, after all). So just how do you create a number in expanded type when it has some spring part?
The structure from the an initial section doesn't change: the expanded form with decimals have to still offer 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 room digits, if bₖ-s and dₖ-s are numbers). Fortunately, we obtain bₖ-s likewise as before; we just have to remember to take the dot into account. To it is in precise, we include as many 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 find dₖ-s by putting as many zeros on the left side that cₖ-s as we have digits between the decimal dot and also the digit in question.
For instance, to uncover d₁, us take c₁ and add as plenty of zeros together we have in between the decimal dot and c₁ (which is, in this case, none). Then, we include the symbols 0. at the really beginning, which offers d₁ = 0.c₁. Similarly, we put one zero come the left that c₂ (since we have one digit between the decimal dot and c₂, namely c₁), and obtain d₂ = 0.0c₂. Us repeat this for every d-s until dₘ = 0.000...cₘ, which has actually m-1 zeros after ~ the decimal dot.
Let's have actually an expanded type example with the number 154.102:
154.102 = 100 + 50 + 4 + 0.1 + 0.002.
(Note exactly how we have actually nothing equivalent to the percentage percent digit. That is since it's same to 0, so in the increased notation, it would be 0.00, or merely 0, i.e., nothing.)
A to crawl eye may have noticed a usual thread once writing number in expanded type (even the expanded kind with decimals): it's all around adding zeros in the ideal places. What is more, zeros normally correspond come 10, 100, 1000, and 0.1, 0.01, 0.001, and so on. An also keener eye might observe that all these numbers space 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 in ~ the expanded kind in math: with exponents.
Expanded type with exponents
Exponents of 10 space very simple. Whenever us take part integer power of 10 (we're no considering fraction exponents here), the an outcome is the number 1 v several zeros that corresponds to the power. As we've seen at the end of the above section, the an initial three positive powers are:
10¹ = 10, 10² = 100, 10³ = 1000,
so the outcomes are the number 1 with one, two, and three zeros, respectively. On the other hand, the very first three an unfavorable powers are:
10⁻¹ = 0.1, 10⁻² = 0.01, 10⁻³ = 0.001,
so again, the number 1 through one, two, and three zeros, respectively, with the slight readjust that the zeros appear to the left rather of right (that's a an outcome of the minus in the exponent).
Another nice home of powers of 10 is that when we multiply any kind of of lock by a one-digit number, the result is the same thing, yet 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 experienced in the broadened notation. In various other words, we can exchange every summand when writing number in expanded kind with a multiplication of something that consists of the number 1 and also some zeros by a one-digit number. And that describes how to create numbers v decimals in expanded type with factors (note just how we can select such an choice in the expanded form calculator).
So what walk expanded kind mean in this case? that again tells us to decompose our numbers right into summands matching to the digits, yet this time, the summands are of the type "digit time a number through 1 and some zeros."
Let's have actually an example to see it clearly. Recall native the over section that:
154.102 = 100 + 50 + 4 + 0.1 + 0.002.
Using the dispute above, we have the right to equivalently create this expanded type example as:
154.102 = 1*100 + 5*10 + 4*1 + 1*0.1 + 2*0.001.
However, we can go also further! Remember exactly how we said at the beginning of this ar that all these components are powers of 10? Well, let's create them together such! This way, we obtain yet one more expanded notation: the expanded type with exponents (observe how we can pick this alternative in the expanded type calculator).
So what is the expanded type with exponents? as before, it's decomposing our number right into summands equivalent to the digits, but now the summands take the form "digit time 10 to part power." In this brand-new variant, the over expanded type example looks like this:
154.102 = 1*10² + 5*10¹ + 4*10⁰ + 1*10⁻¹ + 2*10⁻³.
Observe just how the powers that show up here agree with the subscripts us used in the second section. Also, note how 1 is additionally a power of 10, i.e., the zeroth. In fact, any number raised to power 0 equals 1.
In fact, there's just one thing remaining to do: let's end up with describing exactly how to use the expanded type calculator.
Using the expanded form calculator
The rules governing the expanded form calculator space straightforward. You simply need come follow these three steps:Input the number you'd favor to have actually in increased notation right into the "Number" field.Choose the form you'd favor to have: numbers, factors, or exponents by picking the ideal word in "Show the price in ... Form."Enjoy the result offered to you underneath.
Easy, isn't it? Also, note how for convenience, the expanded kind calculator list consecutive summands row by row and also doesn't cite the terms the correspond to the digits 0 (similarly to how we did as soon as we learned exactly how to create decimals in increased form).
And that's that.
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We've learned the broadened form definition and exactly how to usage it. It's a great starting allude for learning an ext about numbers and how we stand for them, so be certain to examine out Omni's arithmetic calculators ar for more awesome tools.