Expressing Numbers in Standard Form: A Complete Guide
The standard form of numbers is used to simplify and accurately represent very large and very small numbers, making computations easy to communicate efficiently. Because scientific notation makes it possible for scientists to precisely and effectively represent exceedingly very large or small numbers, it is essential to research and astronomy as well as in various other disciplines.
The most basic form of decimal numbers that is easy to read and write is in standard form. Reading and writing extremely small or large numbers can be very difficult at times. We therefore express them in the standard form. The standard form allows for the expression of any number, not only decimals.
In this comprehensive discussion, we will explain the important term of the standard form of numbers. we will elaborate on how to write ordinary numbers in the most famous form which is the standard form.
Defining Standard Form
In the standard form of numbers, we use powers of 10 to express both large and small numbers in a consistent and concise manner. We employ the following formula to express an ordinary number in the standard form:
R x 10k here 1 ≤ R and 10 and k ε ℤ which is the exponent or index (power) of 10.
The standard form of a number is the way we often write them, starting with the greatest place value on the left and moving rightward to the smaller place values. In order to distinguish between hundreds and thousands, in this form place a comma every three numbers from the right.
Note: It is important to note that there are various ways to express the same number using powers of 10 and coefficients, therefore the standard form is not unique.
Representation of Numbers in Standard Form:
The representation of numbers in the standard form is a multiple of a power of 10. Large powers of 10, or positive exponents, are the outcome of the large numbers. Since a number gets smaller when it is multiplied by a decimal, the power of 10 for small integers will be extremely small, signifying a negative exponent.
Use the following important steps to convert a number into standard form:
Step 1: First of all, move the decimal point to the left or right until just one non-zero digit (significant digit) is left. The number that has been formed is the value of R. We can cut off the leading zeros to get 7, for example, changing 7000 into the standard form becomes 7.000.
Step 2: Determine how many times the decimal point was changed. The method will provide a positive number for n if the decimal point is shifted to the left. In the formula, n will have a negative value if the decimal point is moved to the right. Since the decimal point was shifted three times to the left in the example of 7000, k equals three in this case.
Step 3: Using the information obtained from steps 1 and 2, write the number in the form of R x 10k which is the exact standard form of the numbers.
We can elaborate on more precisely:
- Writing Large Numbers:
To write large numbers in standard form, first determine which digits are significant, then count how many decimal places are to the right of the significant digits, and then write the number as R x 10k .
- Writing Small Numbers:
To write large numbers in standard form, first determine which digits are significant, then count how many decimal places are to the left of the significant digits, and then write the number as R x 10-k
How to Write Numbers in Standard Form?
Below are some solved examples to explain how to write in standard form.
Example 1:
What will be the given number 960 000 000 000 000 000 000 in the standard form?
Solution:
Step 1: We can determine the coefficient, which is made up of the non-zero digits (96).
Step 2: After the first non-zero number, such as 9.6, is where the decimal point will be located.
Step 3: Determine what number of digits there are after 9. The decimal point has crossed 20 digits in order to appear in the standard position. This will be 1020 or the exponent of 10.
Step 4: Thus, 9.6 x 1020 . is the standard form representation for the given value.
Example 2:
What will be the given number 0.000 000 000 000 000 000 000 039 in the standard form?
Solution:
Step 1: We can determine the coefficient, which is made up of the non-zero digits (39).
Step 2: After the first non-zero number, such as 3.9, is where the decimal point will be located.
Step 3: Determine what number of digits there are before 3. The decimal point has crossed 23 digits in order to appear in the standard position. This will be 10-23 or the exponent of 10.
Step 4: Thus, 3.9 x 10-23 is the standard form representation for the given value.
Example 3:
What will be the given number 0.000 000 000 000 000 000 000 000 082 in the standard form?
Solution:
Step 1: We can determine the coefficient, which is made up of the non-zero digits (82).
Step 2: After the first non-zero number, such as 8.2, is where the decimal point will be located.
Step 3: Determine what number of digits there are before 8. The decimal point has crossed 26 digits in order to appear in the standard position. This will be 10-26 or the exponent of 10.
Step 4: Thus, 8.2 x 10-26 is the standard form representation for the given value.
Wrap Up:
In this article, we have elaborated on the concept of the standard form of numbers. We explore the representation of the numbers in the standard form precisely. We also discussed the important steps to write the numbers in standard form. In the last section, we solved some examples to understand this concept clearly. We hope that after reading this article you will apprehend the standard form of numbers easily.