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 16th century. This helped in calculations and paved the way for standardized number representation. The modern standard form of a number emerged in the late 19th and early 20th 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 × 106 in standard form is4,500,000
- 4.32 × 10(-5) in standard form is 0.0000432
- 9 × 1011 in standard form is 900,000,000,000
- 2.7 × 10(-10) in standard form is0.00000000027
- 6.8 × 1015 in standard form is6,800,000,000,000,000
Symbolization of standard form
The universal symbolization of a number in standard form is:
⇒ L × 10t
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 × 105. 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 1st 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 × 104 |
| 0.000034 | 3.4 × 10(-5) |
| 5,600,000,000 | 5.6 × 109 |
| 0.0000000027 | 2.7 × 10(-9) |
| 3,750,000,000,000 | 3.75 × 1012 |
| 0.0000000071 | 7.1 × 10(-9) |
| 920,000,000,000,000 | 9.2 x 1014 |
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 × 108 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 × 1013 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 × 109 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 × 105 in standard form.
Solution:
Step 1:
Write the given number.
⇒ 64531284.63 × 105
Step 2:
We check the position of the decimal point.
⇒ 64531284.63 × 105
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 × 105 × 106
⇒ 64531284.63 × 105+6
⇒ 64531284.63 × 1011
Step 5:
Therefore, 64531284.63 × 1011 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 × 1012 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.