It is difficult to write very long Decimal numbers especially if they have to be written multiple times, as in a scientific research paper. For instance, consider numbers like 12345678900000 or 0.000000002345678. To make it easy to read and write these very long numbers, we write them in standard form as 1.23 ✕ 10¹³ and 0.23 ✕ 10¹⁵, respectively. Any number can be written in standard form. In this article, we will learn what is the Standard Form of Numbers, how to express decimal numbers in standard form and how polynomials are written with many solved examples. Read on to learn the standard form of numbers in detail.

Definition of Standard Form of Numbers

It is not easy to read numbers like \({\rm{12983678900000}}\) or \({\rm{0}}{\rm{.0000000021982678}}\). So, when you want to read large and small numbers, you have to write them in the standard form.

Definition: Any number that can be written as the decimal number, between the numbers \({\rm{1}}{\rm{.0}}\) and \({\rm{10}}{\rm{.0}}\), then multiplied by the power of the number \(10\), is known to be in the standard form.

Standard Form of Numbers Example: Write the number \({\rm{81900000000000}}\) in the standard form:

\(81900000000000 = 8.19 \times {10^{13}}\)

We have moved the decimal point \(13\) places to the left side to get the standard form of the number \(8.19 \times {10^{13}}\).

How to Write a Number in Standard Form?

The meaning of the standard form depends on the country you live in.

The standard form is another scientific notation in the United Kingdom and the countries using UK conventions.

\(3890 = 3.89 \times {10^3}\)

In the United States and the countries using the US conventions, the standard form is the usual way of writing numbers in the decimal notation.

The standard form of the number \({\rm{ = 3890}}\)

The expanded form of the above number is \({\rm{3000 + 800 + 90}}\).

The word form of the above number is Three thousand eight hundred and ninety.

Standard Form of a Polynomial

Consider the given algebraic expressions:

\({x^3} + 7{x^2} – 8x\)

\(4{x^5} + 7{x^4} – 5{x^2} – 10\)

\({y^2} + 7{y^6} – 8y – {y^9}\)

\({z^4} + 7{z^5} – 8{z^2} – 12x + 45\)

Can you find out the correlation between the given expressions?

They all belong to the Polynomial family.

The first two equations are written in the standard form in the above examples, whereas the last two are not in their standard form.

The rules for writing the polynomial in the standard form are very straightforward.

1. You have to write the terms in the descending order of their powers of the variables, also known as exponents.

2. You have to make sure that the polynomial has no like terms.

Now, the standard form is:

\({a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} \ldots + {a_0}\)

Thus, the standard form of \({y^2} + 7{y^6} – 8y – {y^9} = – {y^9} + 7{y^6} + {y^2} – 8y\)

The standard form of: \({z^4} + 7{z^5} – 8{z^2} – 12z + 45 = 7{z^5} + {z^4} – 8{z^2} – 12z + 45\)

Standard Form of Rational Number

A rational number \(\frac{p}{q}\) is in its standard form if the denominator \(q\) is the positive and both the integers \(p\) and \(q\) have no common divisor other than \(1\).

You can see the steps which are given to convert the rational number in the standard form:

1. First, you have to write the given reasonable number.

2. Then check if the denominator is positive or negative. If negative, you have to multiply both the numerator and the denominator by \({\rm{ – 1}}\) to make the denominator positive.

3. Now identify the greatest common divisor (GCD) of the numerator and the denominator, which could be dropped.

4. Divide the numerator and the denominator by the GCD.

5. Finally, the obtained number is the standard form of the given rational number.

Expressing Large Numbers in Standard Form

You know that any number between the numbers \(1\) and \(10\) can be expressed as the decimal number by multiplying by a power of \(10\). This form of representation is known as the standard form.

Example: \(500 = 5 \times {10^2}\)

\(587 = 5.87 \times {10^2}\)

\(65.87 = 6.587 \times {10^1}\)

Numbers can be significant as the earth’s diameter or small as the cells present in the human body. So to display these numbers, you can use the exponential form to read and write the numbers quickly.

Example 1: Write the number \(1569000000\;{\rm{km}}\) in standard form.

\(1569000000\;{\rm{km}} = 1.569 \times {10^9}\;{\rm{km}}\)

Example 2: Write the number \(0.000015\;{\rm{m}}\) in standard form.

\(0.000015 = 1.5 \times {10^{ – 5}}\;{\rm{m}}\)

Standard Form of Numbers Facts

1. The number can be known it is in the standard form when the number is written as \(k \times {10^n}\), in this \(1 \le k \le 10\) and \(n\) is the integer.

2. The number can be expressed as the number \(1\) and \(10\) and the integral power of the number \(10\).

3. Standard form is the way of writing the large numbers and the small numbers quickly.

Example: \({10^3} = 1000\), so \(6 \times {10^3} = 6000\).

You can write this in standard form as \(6 \times {10^3}\).

4. The numbers that can be written as decimal numbers between the number \({\rm{1}}{\rm{.0}}\) and \({\rm{10}}{\rm{.0}}\), multiplied by \(10\), are known as the number in standard form.

Solved Examples – Standard Form of Numbers

Let us understand the concept through some Standard Form of Numbers practices problem.

Q.1. Write the given polynomial \({y^2} – 10y + 16 – {y^2} + {y^5} – 3{y^4} + 3{y^2}\) In the standard form.
Ans: To write the given equation in the standard form, we have to consider the below rules:
We have to write the given term in decreasing order of their powers of the variables.
Secondly, all the terms must be unlike.
Now, arrange the given equation in decreasing order:

\({y^2} – 10y + 16 – {y^2} + {y^5} – 3{y^4} + 3{y^2} = {y^5} – 3{y^4} + {y^2} – {y^2} + 3{y^2} – 10y + 16\)

Here, after the addition of the like terms we get,

\({y^5} – 3{y^4} + 3{y^2} – 10y + 16\)

Hence, the standard form of \({y^2} – 10y + 16 – {y^2} + {y^5} – 3{y^4} + 3{y^2}\) is \({y^5} – 3{y^4} + 3{y^2} – {\rm{10 y + 16}}\).

Q.2. Write the standard form for \(3253\)
Ans: Given, \(3253\)
So write the number in decimal form \(3.253 \times 1000\)
Hence, the standard form of the number \(3253\) is \(3.253 \times {10^3}\).

Q.3. Write the number \({\rm{62500000}}\) in the standard form.
Ans: Given, \({\rm{62500000}}\)
We will write the first three digits.
Then by introducing the decimal, write the remaining digits except for the zeroes at the end \({\rm{6}}{\rm{.25}}\).
Now, you have to count the number of digits after the initial digit and then multiply \(10\) to the power of the number \(6.25 \times {10^7}\).
Hence, the required answer is \(6.25 \times {10^7}\).

Q.4. Multiply the numbers \(85000\) and \(2000\) and write the answer in standard form.
Ans: First, write the given numbers in standard form:
Now, multiply the numbers \(85000 = 8.5 \times {10^4}\)
Hence, the required answer is \(1.7 \times {10^8}\).

Q.5. Write the given number \({\rm{ 0}}{\rm{. }}0000012\) in standard form:
Ans: Given, \({\rm{ 0}}{\rm{. }}0000012\)
The standard form of the number \({\rm{ 0}}{\rm{. }}0000012\) is \(1.2 \times {10^{ – 6}}\).
This is \({10^{ – 6}}\) as we have moved the decimal point with six places to the right side to get the number \({\rm{1}}{\rm{.2}}\).

Q.6. Write the given polynomial in standard form: \({x^2} – 10x + 16 – {x^2} + {x^5} – 3{x^4} + 3{x^2}\).
Ans: To write the given equation in the standard form, we have to consider the below rules:
We have to write the given term in decreasing order of their powers.
Secondly, all the terms must be unlike.
Now, arrange the given equation in decreasing order:
\({x^2} – 10x + 16 – {x^2} + {x^5} – 3{x^4} + 3{x^2} = {x^5} – 3{x^4} + {x^2} – {x^2} + 3{x^2} – 10x + 16\)
Here, after the addition of the like terms we get,
\({x^5} – 3{x^4} + 3{x^2} – 10x + 16\)
Hence, the standard form of \({x^2} – 10x + 16 – {x^2} + {x^5} – 3{x^4} + 3{x^2}\)
is \({x^5} – 3{x^4} + 3{x^2} – 10x + 16\).

Summary

In the given article, we have discussed the standard form with an example. Then, we discussed how to write the numbers in standard form followed by the standard form of the polynomial, rational numbers, and large numbers in standard form. Finally, we had glanced at the facts of the standard format. We also solved some examples to understand the concept.

Frequently Asked Questions – Standard Form of Numbers

The answers to the most commonly asked questions on Standard Form of Numbers are given below:

Q.1. Explain the Standard Form of Numbers with an example? Ans: Any number that you can write as the decimal number, between the numbers \({\rm{1}}{\rm{.0}}\) and \({\rm{10}}{\rm{.0}}\), then multiplied by the power of the number \(10\), is known to be in the standard form. Examples: Write the number \(81\,900\,000\,000\,000\) in the standard form: \(81\,900\,000\,000\,000 = 8.19 \times {10^{13}}\) Now, the number is written as \({10^{13}}\). We have moved the decimal point \(13\) places to the left side to get the standard form of the number \({\rm{8}}{\rm{.19}}\).

Q.2. How do you find standard form? Ans: Observe the given steps to find the standard form of the number. Example: Write the standard form of \({\rm{0}}{\rm{.0009}}\). 1. Identify the decimal point in the given number. The decimal point is after the four digits from the left side. 2. Move the decimal point to the first non-zero digit in the given number. 3. Here, you will get \(9\), as there is no non-zero digit after the number \(9\), so there is no need to write the decimal point after \(9\). 4. Count the total number of digits you have moved the decimal point, which is four places further. 5. Now multiply the number with \(10\) and then raise the power of \(10\) with the total number of digits the decimal has been moved. The decimal point has been moved from left to right so that the power will be negative \( \to 9 \times {10^{ – 4}}\). 6. Hence, the standard form of the number \({\rm{0}}{\rm{.0009}}\) is \(9 \times {10^{ – 4}}\).

Q.3. What is a standard form in math? Ans: Standard form is the way of writing the large numbers and the small numbers quickly. Example: \({10^3} = 1000,{\rm{ so }}6 \times {10^3} = 6000\) You can write this in standard form as \(6 \times {10^3}\).

Q.4. What is the standard form of \(450\)? Ans: In general, the standard form of the number \(450\) is the number \(450\) itself – \({\rm{400 + 50 = 450}}\). You can also rewrite the number in scientific notation as \(4.50 \times {10^2}\).

Q.5. How do you write \({\rm{5004300}}\) in standard form? Ans: The given number \({\rm{5004300}}\) can be written in the standard form as shown below: \(5004300 = 5.004300 \times {10^6}\).

We hope this detailed article on the Standard Form of Numbers helps you in your preparation. If you get stuck do let us know in the comments section below and we will get back to you at the earliest.