The standard form of numbers plays a significant role in the study of mathematics and other disciplines. It is especially required in the field of astronomy, to understand complex problems and calculations.
The standard form is the core dimensional concept behind complex calculations. It enables scientists to express long measurements and quantities in a precise and accurate way.
It can also be used in daily life in working with significant figures as well, and it is more common in various disciplines.
In this article, we will discuss how to write numeric in the standard form, and their relevance, explained with some useful examples.
Numbers in Standard Form:
Numbers in standard form which is also known as scientific notation or exponential form is a way to express very large or very small numerals simply and easily.
The general representation of a number in standard form is:
R x 10S
Where
- 1 ≤ R < 10 and sometimes we pronounce it as a coefficient.
- s Є ℤ is the exponent of 10, and it can be both positive and negative.
The standard form plays a pivotal role in dealing with astronomical numbers and microscopic measurements.
Understanding this concept becomes necessary to apprehend the distance between celestial bodies such as stars or galaxies.
It is used where astronomical and microscopic values are very high or small accordingly. Therefore, it is associated with scientific notations.
THINGS TO CONSIDER
The coefficient in standard form is limited to one or more digits but less than 10.
Explanation of the Standard Form:
Main Components:
To enhance our understanding of numeric in standard form, let us address its fundamental components.
a. Mantissa (Coefficient):
The R in the notation stands for the mantissa or coefficient and 1 ≤ R < 10 and s Є ℤ. It comprises the significant digits of the numbers.
b. Power of Ten (Exponent):
The number of digits the decimal point crosses before getting to its proper position indicates the power of 10.
- If the decimal point moves k places to the left, write the exponent of 10 as 10q.
- If the decimal point moves k places to the right, write the exponent of 10 as 10q.
How to Write Numbers in Standard Notation?
To convert numbers in standard or scientific notation, we use the following steps:
- Identify the Coefficient: Write down the significant digits of the original numeric. This forms the coefficient (a).
- Determine the Exponent: Search for the exponent of 10 that is required to bring the coefficient back to the original numeral. The exponent of 10 will be a positive integer if the decimal point is shifted to the left, and a negative integer if the decimal point is shifted to the right.
- Write the Standard Form: Write the coefficient followed by x 10^ and then the exponent. E.g. if you moved the decimal point 4 places to the right and the significant digits were 1.23. The number in standard notation will be 1.23 x 10-4
Converting Small Numbers:
Let us consider the number 0.0000456. To write this into the standard notation, determine the coefficient (4.56) and search the power of ten by counting the decimal places moved (4 places to the right). The given number in standard notation is 4.56 x 10-5
Converting Large Numbers:
We convert a large number such as 7,300,000, 000 into the standard form by determining the coefficient (7.3) and observing the decimal places that are shifted (9 places to the left). 7.3 x 109 is the standard form.
You can utilize an online standard form converter to write larger and smaller numbers to ease up the calculations.
Examples of Writing Numbers in Standard Form
Example 1:
What will be the standard notation of the number 92,781, 000, 000, 000,000?
Solution:
Step 1: Given information:
Number: 92,781, 000, 000, 000,000 (It is an ordinary number)
Step 2: Write down the significant digits of the given number.
92781
Step 3: Put the decimal point between the first two digits.
9.2781
Step 4: Observe that the decimal point will cross 16 digits from the right to the left when we determine the exponent of 10 that is needed to return the coefficient to its original value, to be in the standard position. This will be the exponent of 10.
9.2781 x 1016
Example 2:
What will be the standard notation of the number 0.000000000003492?
Solution:
Step 1: Given information:
Number: 0.000000000003492
Step 2: Write down the significant digits of the given number.
3492
Step 3: Put the decimal point between the first two digits.
3.492
Step 4: The decimal point will cross 12 digits from the right when we determine the exponent of 10 that is needed to return the coefficient to its original value, to be in the standard position. This will be the exponent of 10.
3.492 x 1012
Example 3:
Mars is 141,700, 000 miles away from the sun, and we can write this large value of the distance between Mars and the sun in the standard form as 1.417 x 10^8 miles.
Wrap Up:
The numbers in standard form are a universal notation that signifies the description of very large or very small numerals. In this article, we covered its basic definition and explained this universal concept.
We also described a useful method to write the numeric in the standard form with solved examples. We hope that by apprehending this article, you can convert complex numbers into their standard form easily.
