# Simplifying Common and Rational Numbers: An Introduction to Standard Form

Standard form in mathematics typically refers to a specific way of writing a number, an equation, and rational numbers depending on the context of the number. The standard form provides a clear and consistent way of representing the mathematical number in question, making it easier to compare, manipulate, and analyze.

Standard form is a useful way of writing numbers, especially when dealing with huge numbers then it converts into a small digit. It makes it easier to compare, add, subtract, and multiply numbers. It provides a standardized way of representing numbers that is easy to understand and interpret the result.

In standard form, a number is written as a series of digits with one non-zero digit before the decimal point and then some non-zero number after the decimal point or zero lies between the numbers. i.e., the numbers 123.456 and 12.302 in standard form would be written as 1.23456 x 102 and 1.2302 x 101 respectively.

In this article, we will discuss the definition of standard form, how to represent it, and the steps to write the standard form of the numbers. Moreover, the standard form of rational numbers and solving the examples of a standard form for better understanding.

## Definition and Formula of Standard Form:

When a number is written in digits and powers of 10, it is said to be in its standard form. In standard form, a number is written as a coefficient multiplied by a power of 10.

Standard form = d × 10 n

A coefficient “d” is always a number between 1 and 10, and the power of 10 is “n” which an integer (positive or negative) is. The position of the decimal point in the number is indicated by the power of 10. The power of 10 is a positive or negative number depending on the position of the decimal point that lies in the given number.

For example:

·        The number 4,000 in standard form is written as 4 x 103. The coefficient is 4, between 1 and 10, and the power of 10 is 3, indicating that the decimal point is three places to
the right of the leftmost digit.

·         If the number 0.00005 and the standard form is 5 x 10-5. The coefficient is 5, between 1 and 10, and the power of 10 is -5, indicating that the decimal point is five places to the left of the rightmost digit.

Alternatively

A standard form calculator by Meracalculator can be used to evaluate the results easily when dealing with larger or smaller numbers.

## How to identify the standard form of a number:

In this section, explain the general steps to obtain a number’s standard form:

• Determine the coefficient: The coefficient is the first non-zero digit in the number. If the number is less than 1, count the number of zeros between the decimal point and the first non-zero digit, and use that to determine the coefficient
• Determine the exponent of 10: Count the number of places need to move the decimal point to get to the desired form of the number. The decimal point should move from right to left then the exponent is positive. On the other hand, the exponent is negative if the decimal point swings from the left to the right.
• Write the number in standard form: Write the coefficient followed by the multiplication sign and then raised it to the exponent 10 that is determined in step 2If the exponent is negative, use the appropriate sign.
• Simplify the standard form (if needed): If the coefficient is greater than 10, divide it by 10 and increase the exponent of 10 by 1 until the coefficient is between 1 and 10.

Example:

Convert the 3500000 into the standard form.

Solution:

Step 1: Find the coefficient of the Standard form.

The coefficient is 3.5, Where it is the first non-zero digit in the number.

Step 2: Find the exponent of 10 by determining the position.

We need to move the decimal point six places to the left to get to the desired form of the number, so the exponent is positive 6.

Step 3: The standard form’s formula is written carefully.

Standard form = d × 10 n

Where “d” is the coefficient and “n” is the exponent of 10.

Step 4: Put the values carefully in the above formula.

d = 3.5, n = 6

The standard form of 3500000 = 3.5 x 106

Note:

The coefficient is already between 1 and 10, so no simplification is needed.

## What is the rational number standard form?

A rational number can be expressed as a fraction using the standard form, where the numerator and denominator are both integers and the only common element is “1”.

The numerator and denominator are written in their simplest form, it means that are divided by their greatest common factor. i.e., Rational numbers frequently have the forms 3/4, 5/2, and 7/8. Every term is unique, there is no more common factor in any term except one.

Instruction to write a rational number in standard form:

To generate a rational number into its standard form, follow the instructions below.

·       Reduce the fraction: Divide the numerator and denominator by their largest common factor to minimize the proportion.

·        Check the sign: If the number is negative, write the minus sign in front of the fraction.

·        Form the fraction: The denominator should be written over the numerator.

·        Check for whole numbers: If the numerator and denominator have no common factors other than 1, then the fraction is already in standard form. If the numerator is a multiple of the denominator, then follow the above steps at that time no more common factor of numerator and denominator remained, and the fraction converts into the standard form.

Example:

Construct the standard form of the rational number “24/36”.

Solution:

Step 1: Write the rational number which is given in the question. =24/36

Step 2: Determine the greatest common factor of the above number.

Note, 24 and 36 have “12” as their highest common factor.

Step 3:  Divide the above number in step 1 by their common factor “12” and simplify.

= (24/12) / (36/12)

= 2/3

Noted that, 2 and 3 have no common factors other than 1, so the fraction is in standard form.

The Standard form of the 24/36 is 2/3.

## Summary:

In this article, we discussed the definition and discussed how to find the standard form of numbers. Moreover, discussed the standard form of a rational number and how to find it explained with example. With the whole reading of this article, you can solve the related problem easily.

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