The standard form of a number is a short and standardized way to express extremely large or small numbers. It includes representing a number as the product of a decimal between 1 and 10 and a power of 10. It became the standard way to represent numbers in scientific and technical literature.

Flemish mathematician Simon Stevin introduced decimal notation in the late 16^{th} century. This helped in calculations and paved the way for standardized number representation. The modern standard form of a number emerged in the late 19^{th} and early 20^{th} centuries.

In this article, we will discuss the basic definition of a standard form of a number, its notation, uses, and methods to execute the standard form of a number in detail with examples.

## Standard Form of a Number

The standard form of a number is a way of expressing a number in a brief and standardized format to express very large or very small numbers. It involves representing a number as the product of a decimal between 1 and 10, and a power of 10.

It became widely adopted in scientific and engineering fields, providing a concise and uniform way to express numbers across different magnitudes. It also finds applications in chemistry, where the sizes of atoms and molecules are often stated in this form.

**Illustration:**

- 4.5 × 10
^{6 }in standard form is4,500,000 - 4.32 × 10
^{(-5) }in standard form is 0.0000432 - 9 × 10
^{11 }in standard form is 900,000,000,000 - 2.7 × 10
^{(-10) }in standard form is0.00000000027 - 6.8 × 10
^{15 }in standard form is6,800,000,000,000,000

**Symbolization of standard form**

The universal symbolization of a number in standard form is:

**⇒ ****L × 10 ^{t}**

Where “S” is the number such that **“1≤ S < 10”** and “t” is the integer.

**Example:**

- The number 500,000 in standard form is written as 5 × 10
^{5}. Where A = 5 and n = 5. - The number 0.00027 in standard form is written as 2.7 × 10
^{(-4)}. Where A = 2.7 and n = -4.

## Method to execute the standard form

You can follow the following method.

- Start with the given number that you want to convert to standard form.
- The invention of the 1
^{st}non-zero digit in the number. - If the decimal point is not explicitly shown imagine it just after this non-zero digit.
- Write down the digits after the decimal point if any.
- Tally the number of spaces you stimulated the decimal point to attain the significand from the unique number.
- The total value of the exponent characterizes the number of spaces you moved the fraction point.
- Write the significand (mantissa) as a decimal number between 1 and 10.
- Multiply the significand by 10 raised to the power of the exponent.

**Table of standard form**

Here is a table that demonstrates the standard form of different numbers:

Number | Standard form |
---|---|

10,000 | 1 × 10^{4} |

0.000034 | 3.4 × 10^{(-5)} |

5,600,000,000 | 5.6 × 10^{9} |

0.0000000027 | 2.7 × 10^{(-9)} |

3,750,000,000,000 | 3.75 × 10^{12} |

0.0000000071 | 7.1 × 10^{(-9)} |

920,000,000,000,000 | 9.2 x 10^{14} |

**Note:**

In this table, the left column represents different numbers, and the right column shows their respective standard form. The standard form expresses the numbers as the product of a decimal between 1 and 10 and a power of 10.

This table showcases how numbers of varying magnitudes can be represented in a summarizing and standardized format using scientific notation.

## Daily life uses of Standard Form

The standard form of number has several daily life applications are given below.

**Large Distances:**Standard form is used to represent astronomical distances such as the distance between planets or stars. For instance, the distance between the Earth and the Sun is approximately 1.496 × 10^{8}kilometers.**Small Measurements:**In fields like nanotechnology or microbiology where extremely small measurements are common standard form helps express these values conveniently. For instance, the diameter of a carbon atom is approximately 0.00000000015 meters or 1.5 × 10^{(-10)}meters.**Financial Notation:**Large amounts of money especially in the business world or national debt calculations can be expressed in standard form. For example, a national debt of 22 trillion dollars can be written as 2.2 × 10^{13}dollars.**Population Statistics:**When representing the world population or population growth rates standard form provides a compact representation. For example, the world population of 7.9 billion people can be expressed as 7.9 × 10^{9}people.**Scientific Notation in Chemistry:**In chemistry, scientific notation is used to express the sizes of atoms or molecules. For example, the size of a water molecule is approximately 0.00000000028 meters or 2.8 × 10^{(-10)}meters.**Engineering and Technology:**Standard form is commonly used in engineering and technology fields to express measurements such as voltage, resistance, or frequency. It allows for easier manipulation and comparison of values.**Exponential Growth and Decay:**Standard form is useful for representing exponential growth or decay rates such as in population growth, radioactive decay, or compound interest calculations.

## How to write numbers in Standard Form?

In this section, we have discussed the standard form of numbers with the help of examples.

**Example 1:**

Convert 64531284.63 × 10^{5 }in standard form.

**Solution:**

**Step 1:**

Write the given number.

⇒ 64531284.63 × 10^{5}

**Step 2**:

We check the position of the decimal point.

⇒ 64531284.63 × 10^{5}

**Step 3**:

Fraction point moves the right side of the first non-zero number that is

⇒ 6.453128463

**Step 4**:

Now, count the number of digits you have moved and multiply the number with “raise to power 10”. As we have moved the decimal 7 points to the left side.

⇒ 64531284.63 × 10^{5} × 10^{6}

⇒ 64531284.63 × 10^{5+6}

⇒ 64531284.63 × 10^{11}

Step 5:

Therefore, 64531284.63 × 10^{11} is a standard form.

A standard notation calculator can be used to convert larger and smaller numbers in standard form without any difficulty.

**Example 2:**

Convert 3,750,000,000,000 into standard form.

**Solution:**

**Step 1:**

Write the number.

⇒ 3,750,000,000,000.

**Step 2**:

Check the exact position of the decimal point.

⇒ 3,750,000,000,000.

**Step 3**:

Fraction point moves the right side of the first non-zero number that is

⇒ 3.75

**Step 4**:

Now, count the digits you have moved and multiply the number with “raise to power 10”. As we have moved the decimal 12 points to the left side.

**Step 5:**

Hence, 3.75 × 10^{12} is a standard form.

## Summary

In this article, we have discussed the basic definition of the standard form of a number, the notation of standard form, the table of standard form numbers, daily life uses, and methods to execute the standard form of a number.

In addition, for a better understanding of the standard form of numbers performed different examples. By reading this topic, hope you can solve easily related problems.