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Standard form of numbers is an important concept to represent numbers in scientific notation.Numbers are an integral part of our everyday lives, serving as a fundamental tool for quantifying and expressing various quantities. While working with extremely large or small numbers, it can become burdensome to represent these numbers in their usual form.

The standard form of numbers provides a practical and efficient method of representing extremely large or small quantities. By using a decimal number and a power of 10, it simplifies the communication of such numbers and facilitates comparisons and calculations.

In this article, we will elaborate the idea of standard form of numbers, how to express numbers in standard form, arithmetic operations with numbers in standard form, its merits, applications and how it is applied to express numbers in standard form with examples.

What is Standard Form?

Standard form is a substantial and standardized way of expressing numbers that are either very large or very small. Scientific notation and exponent form are also important terms which are commonly used in place of standard form of numbers.

By expressing these numbers as the product of a decimal number (between 1 and 10) and an exponent of 10,enables us to identify these numbers in a compact manner. The number of places to which the decimal point must be shifted to get the original value is indicated as the exponent of 10. Mathematically,

A x 10^ⁿ, where 1 ≤A< 10 and n is an integer.

Here M is a coefficient and it is a decimal number range from equal or greater than 1 but less than 10. Moreover n (power of ten) specifies the scale or order of magnitude.

Steps to Express Numbers in Standard Form:

To express a number in standard form, we follow a specific format.

1.Identify the decimal number between 1 and 10:

Always choose a number that is higher than or equal to one but less than ten.

1. Find out the exponent of 10:

Determine how many places the decimal point has to be moved to get the original value. If decimal point moves from left side or right side, then the exponent of 10 will be negative or positive respectively.

You can try a standard form converter to express numbers in scientific notation according to the above-mentioned steps.

Combine the decimal number and the power of 10:

In this step, you’re to combine the decimal number and the exponent of 10 which you will have identified in the above two steps.

Operations with Numbers in Standard Form:

Performing arithmetic operations with numbers in standard form is simple. Here's how you can carry out basic operations:

Addition and Subtraction:

• Ensure that both numbers have the same power of 10.
• Add or subtract the decimal numbers.
• Keep the common power of 10.
• Write the answer in scientific notation.

Multiplication:

• Multiply the decimal numbers.
• Add the powers of 10.
• Adjust the result to ensure it remains in standard form if necessary.

Division:

• Divide the decimal numbers.

• Subtract the powers of 10.

• Adjust the result to ensure it remains in standard form if necessary.

Advantages of Standard Form:

The standard form of numbers offers several advantages in representing and comparing numbers that differ greatly in magnitude. Here are a few benefits:

Concise representation:

Standard form of numbersenables us to write extremely large or small numbers in a concise and easily understandable format, making it convenient for mathematical calculations and scientific notation.

Easy comparison:

By representing numbers in standard form, it becomes effortless to compare their magnitudes. The power of 10 provides a clear indication of which number is larger or smaller.

Simplified arithmetic:

It is convenient and simple to use arithmetic operations (addition, subtraction, multiplication, and division) when working with numbers that are written in standard form. Additionally, it reduces one's need for handling very small or massive numbers.

Compatibility with scientific notation:

Standard form of numbers is closely related to scientific notation, which is widely used in scientific research, engineering, and mathematics. Familiarity with standard form enhances understanding and communication within these fields.

Applications of Standard Form:

Standard form finds application in various scientific and mathematical disciplines. Some common areas where standard form is frequently employed include:

Astronomy:

Representing astronomical distances, masses, and quantities.

Physics:

Expressing the values of fundamental physical constants, such as the speed of light or Planck's constant.

Chemistry:

Describing molecular masses, atomic sizes, and concentrations.

Economics:

Presenting large financial figures or GDP values.

Scientific Notation:

Scientists often employ standard form to express measurements, such as distances between celestial objects, molecular sizes, or energy values.

Engineering Applications:

Engineers utilize standard form to represent quantities like voltage, resistance, and power in electrical circuits, as well as dimensions and quantities in structural analysis.

Financial Reports:

Standard form facilitates the representation of large financial figures in reports, such as company revenues, national debt, or stock market values.

Medical Science:

In medical research and practice, standard form is employed to express quantities like drug concentrations, cell counts, or molecular weights.

How to express numbers in standard notation?

Example 1:

Express the ordinary number 738000000000000000000 in standard form.

Solution:

Step 1: Write the given number

738000000000000000000

Step 2: We identify the decimal number that is 738 and standard position is after the first decimal number i.e. 7.38.

Step 3: As decimal point is shifted 20 places from right side. Thus, exponent of 10 will be +20. Hence number in standard form is 7.38 x 10^{^20}.

Example 2:

Express the given ordinary number 0.000000000000529 in standard form of numbers.

Solution:

Step 1: Write the given number

0.000000000000529

Step 2: We identify the decimal number which is 428 and standard position is after the first decimal number i.e. 5.29

Step 3: As the decimal point is shifted places from left side. Thus, exponent of 10 will be -13. Hence number in standard form is 5.29 x 10^{^ -13}.

Conclusion:

In this article, we have discussed standard form of numbers in detail. We have elaborated the way to write ordinary numbers in standard form, arithmetic operations, benefits, applications as well as some examples. Hopefully, reading this article you will be able to apprehend the concept of standard form of numbers.