Welcome come Omni's expanded form calculator - your article of selection for learning how to compose numbers in increased form. In essence, the expanded kind in mathematics (also called expanded notation) is a way to decompose a value into summands matching to its digits. The object is comparable to scientific notation, though here, we break-up it right into even much more terms. To do the link even clearer, we have three different options of writing numbers in expanded type in the calculator, such as the expanded form with exponents.
You are watching: Expanded form using powers of 10
Expanded type is critical in miscellaneous parts that math, e.g., in partial assets algorithm. So what is expanded form? Well, let's jump straight into the article and find out!
What is the increased form?
The expanded kind definition is the following:
|💡 creating numbers in expanded kind means reflecting the value of every digit. To it is in precise, us express the number as a sum of terms the correspond to the digit of ones, tens, hundreds, etc., as well as those that tenths, hundredths, and also so on for the expanded kind with decimals.|
As discussed above, the increased notation of, say, 154 need to be a sum of terms, each connected to one of the digits. Obviously, we can't simply write 1 + 5 + 4 since that's miles away from what us had. So exactly how do you compose a number in expanded form? Well, you add zeros.
154 = 100 + 50 + 4
So what walk expanded kind mean? Intuitively, us associate every digit of the number v something that has actually the exact same digit, followed by sufficiently plenty of zeros to finish up in the ideal position as soon as we amount it all up. To make it more precise, let's have it neatly described in a different section.
How to compose numbers in expanded form
Let's take a number that has the form aₙ...a₄a₃a₂a₁a₀, i.e., the aₖ-s denote continuous digits of the number through a₀ being the people digit, a₁ the tens digit, and so on. According to the broadened form meaning from the previous section, we'd like 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 form starting native the appropriate side, i.e., native a₀. Due to the fact that it is the ones' digit, it must appear at the end of our number. We produce b₀ by writing as numerous zeros to the appropriate of a₀ as we have digits after a₀ in our number. In various other words, we add none and get b₀ = a₀.
Next, we have actually the 10s digit a₁. Again, we type b₁ by putting as plenty of zeros to the ideal of a₁ as we have digits following a₁ in the initial number. In this case, there's one such (namely, a₀), for this reason we have b₁ = a₁0 (remember that below we use the notation of composing digit ~ digit). Similarly, to b₂ we'll add two zeros (since a₂ has a₁ and a₀ to the right), definition that b₂ = a₂00, and so on till bₙ = aₙ00...000 through n-1 zeros.
Alright, we've seen how to write numbers in expanded type in a special instance - when they're integers. However what if we have decimals? Or if it's part long-expression with several numbers before and also after the dot? What is the expanded type of together a monstrosity?
Well, let's see, candlestick we?
How to write decimals in expanded form
Essentially, we carry out the same as in the above section. In short, us again add a suitable number of zeros come a digit however for those ~ the decimal dot, we write them to the left rather of to the right. Obviously, the dot should be placed at the appropriate spot so that it all makes sense (we can't have an integer starting with zeros, after all). So how do you create a number in expanded type when it has some spring part?
The framework from the first section doesn't change: the expanded type with decimals need to 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 also cₖ-s are digits, while bₖ-s and dₖ-s room numbers). Fortunately, we attain bₖ-s similarly as before; us just have to remember come take the dot right into account. To be precise, we add as plenty of zeros together we have digits to the right, but before the decimal dot (i.e., we only count the a-s).
On the other hand, we discover dₖ-s by placing 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 include as many zeros together 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 in between the decimal dot and c₂, namely c₁), and also obtain d₂ = 0.0c₂. Us repeat this for every d-s till dₘ = 0.000...cₘ, which has actually m-1 zeros after the decimal dot.
Let's have actually an expanded form example with the number 154.102:
154.102 = 100 + 50 + 4 + 0.1 + 0.002.
(Note just how we have actually nothing equivalent to the percentage percent digit. The is because it's equal to 0, for this reason in the expanded notation, it would certainly be 0.00, or just 0, i.e., nothing.)
A crawl eye may have noticed a usual thread when writing number in expanded form (even the expanded form with decimals): it's all around adding zeros in the appropriate places. What is more, zeros normally correspond to 10, 100, 1000, and also 0.1, 0.01, 0.001, and so on. An even keener eye might observe the all this 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 united state to a new means of looking in ~ the expanded type in math: with exponents.
Expanded kind with exponents
Exponents the 10 space very simple. Whenever we take some integer strength of 10 (we're no considering fraction exponents here), the result is the digit 1 with several zeros that coincides to that power. Together we've seen at the finish of the over section, the an initial three positive powers are:
10¹ = 10, 10² = 100, 10³ = 1000,
so the results are the number 1 v one, two, and also three zeros, respectively. Top top the other hand, the very first three negative powers are:
10⁻¹ = 0.1, 10⁻² = 0.01, 10⁻³ = 0.001,
so again, the number 1 v one, two, and also three zeros, respectively, with 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 residential property of powers of 10 is that once we multiply any kind of of lock by a one-digit number, the result is the very 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 as with the summands we observed in the expanded notation. In other words, we might exchange every summand when writing numbers in expanded type with a multiplication of other that is composed of the digit 1 and some zeros by a one-digit number. And also that explains how to compose numbers through decimals in expanded form with factors (note just how we can select such an alternative in the expanded kind calculator).
So what walk expanded form mean in this case? it again tells united state to decompose ours numbers right into summands corresponding to the digits, however this time, the summands space of the kind "digit time a number through 1 and some zeros."
Let's have actually an example to check out it clearly. Recall native the over section that:
154.102 = 100 + 50 + 4 + 0.1 + 0.002.
Using the argument above, we have the right to 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 also further! Remember how we said at the beginning of this section that every these components are strength of 10? Well, let's write them together such! This way, we acquire yet one more expanded notation: the expanded form with exponents (observe how we can select this choice in the expanded type calculator).
So what is the expanded kind with exponents? together before, it's decomposing our number into summands matching to the digits, but now the summands take it the type "digit times 10 to some 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 v the subscripts us used in the second section. Also, note just how 1 is likewise a strength of 10, i.e., the zeroth. In fact, any number raised to power 0 equals 1.
In fact, there's only one thing staying to do: let's finish with describing exactly how to use the expanded kind calculator.
Using the expanded type calculator
The rules governing the expanded type calculator are straightforward. You just need come follow these 3 steps:Input the number you'd favor to have in increased 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 price in ... Form."Enjoy the result provided to you underneath.
Easy, isn't it? Also, note just how for convenience, the expanded type calculator lists consecutive summands row by row and doesn't mention the terms that correspond to the digits 0 (similarly to how we did once we learned exactly how to write decimals in broadened form).
And that's that.
See more: How Much Does 8.5X11 Paper Weigh ? How Much Does 20
We've learned the increased form meaning and how to use it. It's a an excellent starting suggest for learning much more about numbers and how we represent them, so be sure to inspect out Omni's arithmetic calculators section for more amazing tools.