Understanding Standard Form of Numbers: A Comprehensive Guide

The standard form of numbers is a simple way to write big or small numbers using digits and powers of ten. It makes numbers easier to read and compare. It helps in the easy representation of large numbers. This method doesn’t have a single inventor but evolved to simplify calculations. 

The standard form of numbers is understood differently in various regions. For instance, in the United Kingdom and places following UK conventions, the standard form refers to what is otherwise termed Scientific notation for numbers.

In this article, we’ll explore the core ideas behind expressing numbers using the standard format. We’ll also solve some examples to gain more insight into the standard form of numbers.

Definition of Standard Form:

Standard form and scientific notation are two names for the same thing. It is a method used in mathematics to represent very large or tiny numbers briefly. It involves expressing a number as a product of a decimal between 1 and 10 multiplied by a power of 10. 

This form simplifies the representation of numbers and makes it easier to work with extremely large or tiny values in calculations or scientific notation.

Formula of Standard Form:

The formula for expressing a number in standard form is:

Number = Coefficient * 10ⁿ

Where:

• A decimal value less than 10 and larger than or equal to 1 is the coefficient.
• n represents the exponent or power of 10 required to reach the original number’s value.

Procedure to write the Standard Form of a Number:

Below are a few steps to write the standard form of numbers in the following manner:

1. Identify the first digit we want to express in standard form.

2. Place the decimal point immediately following the first digit.

3. Count the places we moved the decimal point from its original position to get the new decimal value.

4. This count becomes the exponent or power of 10.

Procedures for Writing Decimal Numbers in Standard Notation:

We use a base of 10 and 10 different symbols along with a dot to represent values in decimal numbers. Each digit holds a specific value based on its position within the number. 

For Example:

789.432

• 7 hundreds, 8 tens, 9 ones, 4 tenths, 3 hundredths, and 2 thousandths.

1. Identify the first non-zero digit from the left.
2. Write this digit as the coefficient in decimal form, placing the decimal point after it.
3. Count the number of decimal places between this digit and the original decimal point to determine the exponent for the power of 10.

Procedure to Write Rational Numbers in Standard Form:

A fraction expressed as p/q is considered to be in standard form when the denominator q is a positive number and both the numerator p and the denominator q do not share any common divisors except for 1.

Steps:

1. Express the provided rational number.
2. Determine the sign of the denominator. If negative, multiply both numerator and denominator by -1 to obtain a positive denominator.
3. Identify the greatest common factor (GCD) of the numerator and denominator. The GCD is the largest integer that divides both the numerator and denominator evenly.
4. Use the GCD to divide both the top (numerator) and bottom (Denominator) numbers.
5. The resulting fraction represents the given rational number in its standard form. 

How to Write Decimals in Their Expanded Forms?

Decimals, like whole numbers, can be written in different representations. One such representation is the expanded form, which breaks down the decimal into its constituent place values. 

Here’s an example to make clear. 

Example: Show the given number 4.725 in expanded form.

4.725 = 4 * 10⁰ + 7 * 10 ^-1 + 2 * 10^-2 + 5 * 10^-3

= 4 * 1 + 7 * 0.1 + 2 * 0.01 + 5 * 0.001

 = 4 + 0.7 + 0.02 + 0.05

Solved Examples of Standard Form:

Example 1: Writing a Large Number in Standard Form

Number: 98,000,000 

Solution:

1. Identify the first digit: 9.
2. Decimal point should be placed after 9: 9.8
3. Count the number of places moved: Moved 7 places to the left.
4. Express in standard form: 9.8×10⁷

Example 2: Writing a Small Number in Standard Form

Number: 0.0000456 

Solution:

1. Identify the first non-zero digit: 4.
2. Write this digit as the coefficient: 4.56
3. Count the number of decimal places: Moved 4 places to the right.
4. Express in standard form: 4.56×10⁻⁴

Example 3: Converting Rational Number to Standard Form

Rational Number: 18 /25​ 

Solution:

1. Express the provided rational number.
2. The denominator is already positive.
3. Find the greatest common divisor (GCD) of 18 and 25: GCD = 1.
4. Divide both the upper and lower number by the GCD: 18÷1/ 25÷1= 18 / 25
5. The standard form remains: 18 / 25

Example 4: Large Decimal Number in Standard Form

Decimal Number: 672.5489 

Solution:

• Place value breakdown: 6 hundred, 7 tens, 2 ones, 5 tenths, 4 hundredths, 8 thousandths, and 9 ten-thousandths.
• Write the number in standard form: 6.725489×1026.725489×102

Wrap Up:

In this article, we explored the concept of standard form for numbers. We learned its definition, formula, and steps for writing different kinds of numbers (decimal, large, small, and rational) in standard form, and saw some solved examples. We also discovered its practical applications in various fields like science, physics, engineering, and finance.