The standard form is a method of expressing numbers that makes them easier to understand. It is a helpful tool for dealing with extremely big and very small numbers. It uses a combination of digits and powers (exponent) of ten to represent a number.
The standard notation helps scientists and mathematicians to perform calculations and comparisons more effectively. This saves time and effort to write down large numbers of zeros. For example; the mass of the Sun is approximately 1989000000000000000000000000000 kilograms. This can be written in standard form as 1.989 x 1030 kilograms.
We will explore the definition of the standard form of a number with its mathematical representation in this article. We will learn how to express numbers in their standard form with the help of numerous examples.
Definition and Explanation of the Standard Form
The standard form is a way of writing down very large or extremely small numbers using powers of 10. It is a method of expressing numbers as a single digit followed by a decimal point and then a series of zeroes, multiplied by an exponent of 10. In the United Kingdom, it is also known as scientific form or standard index form.
In standard notation, non-zero numbers are written in the form
a × 10n
- ‘a’ is a decimal number larger than or equal to 1 and smaller than 10. It is referred to as the coefficient and it represents the significant digits of the number.
- ‘n’ is an integer (positive or negative) that represents the exponent of 10. It indicates the order of magnitude of the number and tells how many places to move the decimal point to get the original number.
Converting Whole Numbers to Standard Form
Here is a systematic guide on how to convert numbers to standard form:
- Start by writing the first digit from the left in the given whole number.
- Place the decimal point immediately after this digit.
- Count how many digits are there that appear after the first digit. This count will determine the exponent of 10 in the standard form representation.
Let’s look at an example to explain the preceding steps:
Example: Convert the whole number 5250000 into standard form.
Step 1: Write the first digit, which is 5.
Step 2: Insert a point after 5 (i.e. 5.).
Step 3: Count the number of digits following the first digit. There are 6 digits after 3.
Therefore, in standard form, the number 5250000 is represented as 5.25 × 106.
You can also use the online standard form calculator by MeraCalculator to quickly convert large numbers into standard notation.
Converting Decimal Numbers to Standard Form
Here is a step-by-step explanation of how to write decimal numbers in standard notation.
- Write the first non-zero digit of the provided number.
- Put the decimal point right after that digit.
- Count how many times you moved the decimal point to reach the first non-zero digit.
- Show this count as an exponent of 10.
- Use a negative sign on the exponent of 10 if you moved the decimal point to the right.
Example: Convert the decimal number 0.0000000001234 to standard notation.
Step 1: The first nonzero digit is 1.
Step 2: Place the decimal point after 1 (i.e. 1.).
Step 3: The decimal point needs to move 10 places to the right to obtain the original number.
Step 4: Since the decimal point moves to the right, we write a negative sign on the exponent of 10.
Thus, the standard form of 0.0000000001234 is 1.234 × 10-10.
The Conversion from Standard Form to Ordinary Numbers
Here are the steps to convert a number from standard form to an ordinary number:
- Look at the standard form (a × 10n) number and identify the coefficient (a) and the exponent (n).
- If the exponent is positive, the number is large. The number is small if it is negative.
- If n is positive, move the decimal point in the coefficient a to the right by n positions. Add zeros as needed if you run out of digits. This effectively increases the size of the number.
- If n is negative, move the decimal point in the coefficient a to the left by n positions.
- After moving the decimal point, write down the resulting number without the × 10n part. This is the ordinary number.
Example: Convert 6.3 × 104 to an ordinary form.
Step 1: Identify the Coefficient (6.3) and Exponent (4).
Step 2: The exponent is positive (4), indicating a large number.
Step 3: Move the decimal point in 6.3 four positions to the right: 63000.
Hence, 63000 is the ordinary of 6.3 × 104.
Solved Examples of Standard Form/Scientific Notation
Here are some examples of standard form or scientific notation with Solution:
Write the number 6500000 in standard form.
6500000 can be expressed in scientific notation as 6.5 x 106.
Write the number 0.000025 in standard form.
0.000025 can be expressed in scientific notation as 2.5 x 10-5.
Express 9.6 x 10-6 in standard form.
9.6 x 10-6 can be written as 0.0000096 in standard form.
Applications of Standard Form in Other Fields
Proficiency in standard numeric notation representation is not limited to mathematics but it also helps to effectively handle both large and small numerical values across various disciplines.
Astronomy: Astronomers use standard forms to express the vast distances between celestial objects.
Chemistry: Chemists use it to represent the size of molecules and atoms.
Economics: Economists use the standard form to express large GDP figures and national debts.
Engineering: Engineers use it when dealing with measurements on different scales, such as the size of microchips or the distance between planets.
We have explored the definition and mathematical representation of standard form, including how to convert both whole numbers and decimal numbers into this notation. We have discussed the conversion from standard form back to ordinary numbers. We have included numerous examples to illustrate the concept of standard form. Remember that practice makes perfect, so keep exploring and expanding your knowledge of standard form.