How To Write 309 017 In Expanded Form

Expanded Grade Calculator

Created by Maciej Kowalski , PhD candidate

Reviewed past

Steven Wooding

Last updated:

Dec 13, 2022

Welcome to Omni's expanded form reckoner - your article of choice for learning how to write numbers in expanded form. In essence, the expanded grade in math (also called expanded annotation) is a way to decompose a value into summands respective to its digits. The topic is like to scientific notation, though here, we divide it into even more terms. To brand the connectedness even clearer, nosotros have three different options of writing numbers in expanded grade in the estimator, such as the expanded grade with exponents.

Expanded form is crucial in diverse parts of math, e.g., in the fractional products algorithm. Then what is expanded class? Well, allow'south jump straight into the article and discover out!

What is the expanded course?

The expanded grade 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., too as those of tenths, hundredths, and so on for the expanded grade with decimals.

As mentioned above, the expanded notation of, say, 154 should exist a sum of terms, each continued to one of the digits. Obviously, nosotros can't but write 1 + 5 + four since that'southward miles away from what we had. So how practice you write a number in expanded form? Well, you add zeros.

154 = 100 + 50 + 4

So what does expanded course hateful? 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 nosotros sum information technology all upwards. To make information technology more than precise, allow's have it neatly described in a separate section.

How to write numbers in expanded form

Let's accept a number that has the grade aₙ...a₄a₃a₂a₁a₀, i.due east., the aₖ-due south denote sequent digits of the number with a₀ being the ones digit, a₁ the tens digit, and so on. According to the expanded course definition from the previous section we'd similar to write:

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

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

Let'due south explain how to write such numbers in expanded form starting from the correct side, i.east., from a₀. Since it is the ones' digit, information technology must appear at the cease of our number. We create b₀ by writing as many zeros to the right of a₀ as we take digits after a₀ in our number. In other words, nosotros add none and get b₀ = a₀.

Next, we have the tens digit a₁. Once again, nosotros form b₁ by putting as many zeros to the right of a₁ as we accept digits following a₁ in the original number. In this case, at that place's ane such (namely, a₀), so we accept b₁ = a₁0 (recall that here we utilise the notation of writing digit after digit). Similarly, to b₂ nosotros'll add two zeros (since a₂ has a₁ and a₀ to the right), pregnant that b₂ = a₂00, and so on until bₙ = aₙ00...000 with n-1 zeros.

Alright, nosotros'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'southward some long-expression with several numbers before and after the dot? What is the expanded form of such a monstrosity?

Well, permit's see, shall we?

How to write decimals in expanded course

Substantially, we do the aforementioned as in the above section. In short, we again add a suitable number of zeros to a digit just for those later on the decimal dot, nosotros write them to the left instead of to the correct. Evidently, 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). Then how practise you write a number in expanded form when it has some fractional office?

The framework from the outset section doesn't alter: the expanded form with decimals should yet requite united states of america 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-southward are digits, while bₖ-s and dₖ-southward are numbers). Fortunately, we obtain bₖ-south similarly as before; nosotros but take to remember to take the dot into account. To be precise, nosotros add equally many zeros every bit we take digits to the right, but earlier the decimal dot (i.e., we merely count the a-due south).

On the other paw, nosotros detect dₖ-s by putting equally many zeros on the left side of cₖ-southward every bit we have digits betwixt the decimal dot and the digit in question.

For instance, to detect d₁, we have c₁ and add as many zeros as we have between the decimal dot and c₁ (which is, in this example, none). And so, we add the symbols 0. at the very beginning, which gives d₁ = 0.c₁. Similarly, nosotros put 1 nix to the left of c₂ (since we take one digit between the decimal dot and c₂, namely c₁), and obtain d₂ = 0.0c₂. We echo this for all d-due south until dₘ = 0.000...cₘ, which has k-1 zeros after the decimal dot.

Let'south take an expanded form example with the number 154.102:

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

(Note how nosotros accept cipher corresponding to the hundredths digit. That is because it'southward equal to 0, so in the expanded notation, it would be 0.00, or simply 0, i.e., nothing.)

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

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

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

Expanded form with exponents

Exponents of 10 are very unproblematic. Whenever we take some integer ability of x (we're not because fraction exponents here), the consequence is the digit one with several zeros that corresponds to that power. Equally we've seen at the end of the above section, the first three positive powers are:

10¹ = 10, 10² = 100, ten³ = 1000,

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

x⁻¹ = 0.ane, ten⁻² = 0.01, 10⁻³ = 0.001,

then again, the digit 1 with one, two, and iii 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).

Some other dainty property of powers of x is that when we multiply any of them by a one-digit number, the result is the same affair, merely with the 1 replaced by that number. For case:

x * 5 = 50, 1000 * iii = 3000, 0.001 * 6 = 0.006,

and these look just like the summands we saw in the expanded notation. In other words, nosotros could exchange every summand when writing numbers in expanded grade with [a multiplication of something that consists of the digit ane 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 cull such an option in the expanded form reckoner).

And so what does expanded grade hateful in this case? It again tells us to decompose our numbers into summands corresponding to the digits, only this time, the summands are of the class "digit times a number with i and some zeros."

Let'south have an instance to see it conspicuously. Recall from the above section that:

154.102 = 100 + fifty + 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.i + two*0.001.

However, we can get fifty-fifty further! Recollect how we said at the first of this section that all these factors are powers of 10? Well, permit's write them as such! This manner, nosotros obtain yet another expanded notation: the expanded course with exponents (detect how we can cull this selection in the expanded form calculator).

So what is the expanded grade with exponents? Every bit before, it's decomposing our number into summands respective to the digits, but now the summands take the form "digit times 10 to some power." In this new variant, the to a higher place expanded course example looks like this:

154.102 = 1*10² + five*x¹ + 4*x⁰ + 1*10⁻¹ + 2*10⁻³.

Observe how the powers that announced hither concord with the subscripts nosotros 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 unlike ways: with numbers, with factors, and with exponents.

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

Using the expanded form estimator

The rules governing the expanded grade estimator are straightforward. You lot simply demand to follow these three steps:

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

Easy, isn't it? Likewise, note how for convenience, the expanded class 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 nosotros learned how to write decimals in expanded course).

And that's that. We've learned the expanded form definition and how to utilise it.

Maciej Kowalski , PhD candidate

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Source: https://www.omnicalculator.com/math/expanded-form

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