Scientific notation is a method for expressing a given quantity as a number having significant digits necessary for a specified degree of accuracy, multiplied by 10 to the appropriate power such as 1.56 × 107. It is a form of presenting very large numbers or very small numbers in a simpler form. The scientific notation helps us to represent the numbers that are very huge or very tiny in a form of multiplication of single-digit numbers and 10 raised to the power of the respective exponent. The exponent is positive if the number is very large and it is negative if the number is very small. Let’s understand the scientific notation formula.
What is Scientific Notation Formula?
A scientific notation is a form of writing a given number, an equation, or an expression in a form that follows certain rules. Writing a large number like 8.6 billion in its number form is not just ambiguous but also time-consuming and there are chances that we may write a few zeros less or more while writing in the number form. So, to represent very large or very small numbers concisely, we use scientific notation. The general representation of scientific notation or scientific notation formula is given as:
a × 10b ; 1 ≤ a < 10
‘a’ represents any number between 0 to 10 ( greater than 0 and lesser than 10)where,
Scientific Notation Rules
To determine the power or exponent of 10, let us understand how many places we need to move the decimal point after the single-digit number.
- If the given number is multiples of 10 then the decimal point has to move to the left, and the power of 10 will be positive.
Example: 8000 = 8 × 103 is in scientific notation.
- If the given number is smaller than 1, then the decimal point has to move to the right, so the power of 10 will be negative.
Example: 0.0005 = 5 × 0.0001 = 6 × 10-4 is in scientific notation.
Positive and Negative Exponent
- When the scientific notation of any large numbers is expressed, then we use positive exponents for base 10.
Example: 5000000 = 5 x 106, where 6 is the positive exponent.
- When the scientific notation of any small numbers is expressed, then we use negative exponents for base 10.
Example: 0.00004 = 4 x 10-5, where -5 is the negative exponent.
Let us see the applications of the scientific notation formula in the following solved examples.
Solved Examples Using Scientific Notation Formula
Example 1: Convert 0.00000084 into scientific notation using scientific notation formula.
Solution:
Move the decimal point to the right of 0.000000084 up to 8 places.
The decimal point was moved 8 places to the right to form the number 8.4
Since the numbers are less than 10 and the decimal is moved to the right, so we use a negative exponent here.
⇒ 0.000000084 = 8.4 × 10-8
Answer: Thus, in scientific notation 0.000000084 is expressed as 8.4 × 10-8
Example 2: Can you help Zoe to write 7.56×1011
in the standard notation?
Solution
Here, 7.56 is 756
. Now, Zoe has to move the decimal point 11 places to the right and add zeroes accordingly.
The standard notation for 7.56×1011 is 756,000,000,000
Answer: Thus, the standard form is 756,000,000,000.
Scientific Notation
Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers:
It makes it easy to use big and small values.
OK, How Does it Work?
Example: 700
Why is 700 written as 7 × 102 in Scientific Notation ?
700 = 7 × 100
and 100 = 102(see powers of 10)
so 700 = 7 × 102
Both 700 and 7 × 102 have the same value, just shown in different ways.
Example: 4,900,000,000
1,000,000,000 = 109 ,
so 4,900,000,000 = 4.9 × 109 in Scientific Notation
The number is written in two parts:
- Just the digits, with the decimal point placed after the first digit, followed by
- × 10 to a power that puts the decimal point where it should be
(i.e. it shows how many places to move the decimal point).
Other Ways of Writing It
3.1 × 10^8
We can use the ^ symbol (above the 6 on a keyboard), as it is easy to type.
Example: 3 × 10^4 is the same as 3 × 104
- 3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000
Calculators often use “E” or “e” like this:
Example: 6E+5 is the same as 6 × 105
- 6E+5 = 6 × 10 × 10 × 10 × 10 × 10 = 600,000
Example: 3.12E4 is the same as 3.12 × 104
- 3.12E4 = 3.12 × 10 × 10 × 10 × 10 = 31,200
How to Do it
To figure out the power of 10, think “how many places do I move the decimal point?”
When the number is 10 or greater, the decimal point has to move to the left, and the power of 10 is positive.
When the number is smaller than 1, the decimal point has to move to the right, so the power of 10 is negative.
Example: 0.0055 is written 5.5 × 10-3
Because 0.0055 = 5.5 × 0.001 = 5.5 × 10-3
Example: 3.2 is written 3.2 × 100
We didn’t have to move the decimal point at all, so the power is 100
But it is now in Scientific Notation
Check!
After putting the number in Scientific Notation, just check that:
- The “digits” part is between 1 and 10 (it can be 1, but never 10)
- The “power” part shows exactly how many places to move the decimal point
Why Use It?
Because it makes it easier when dealing with very big or very small numbers, which are common in Scientific and Engineering work.
Example: it is easier to write (and read) 1.3 × 10-9 than 0.0000000013
It can also make calculations easier, as in this example:
Example: a tiny space inside a computer chip has been measured to be 0.00000256m wide, 0.00000014m long and 0.000275m high.
What is its volume?
Let’s first convert the three lengths into scientific notation:
- width: 0.000 002 56m = 2.56×10-6
- length: 0.000 000 14m = 1.4×10-7
- height: 0.000 275m = 2.75×10-4
Then multiply the digits together (ignoring the ×10s):
2.56 × 1.4 × 2.75 = 9.856
Last, multiply the ×10s:
10-6 × 10-7 × 10-4 = 10-17 (easier than it looks, just add −6, −4 and −7 together)
The result is 9.856×10-17 m3
It is used a lot in Science:
Example: Suns, Moons and Planets
The Sun has a Mass of 1.988 × 1030 kg.
Easier than writing 1,988,000,000,000,000,000,000,000,000,000 kg
(and that number gives a false sense of many digits of accuracy.)
It can also save space! Here is what happens when you double on each square of a chess board:
Values are rounded off, so 53,6870,912 is shown as just 5×108
That last value, shown as 9×1018 is actually 9,223,372,036,854,775,808
Engineering Notation
Engineering Notation is like Scientific Notation, except that we only use powers of ten that are multiples of 3 (such as 103, 10-3, 1012 etc).
Examples:
- 2,700 is written 2.7 × 103
- 27,000 is written 27 × 103
- 270,000 is written 270 × 103
- 2,700,000 is written 2.7 × 106
Example: 0.00012 is written 120 × 10-6
Notice that the “digits” part can now be between 1 and 1,000 (it can be 1, but never 1,000).
The advantage is that we can replace the ×10s with Metric Numbers. So we can use standard words (such as thousand or million), prefixes (such as kilo, mega) or the symbol (k, M, etc)
Example: 19,300 meters is written 19.3 × 103 m, or 19.3 km
Example: 0.00012 seconds is written 120 × 10-6 s, or 120 microseconds
Scientific Notation Definition
As discussed in the introduction, the scientific notation helps us to represent the numbers which are very huge or very tiny in a form of multiplication of single-digit numbers and 10 raised to the power of the respective exponent. The exponent is positive if the number is very large and it is negative if the number is very small. Learn power and exponents for better understanding.
The general representation of scientific notation is:
| a × 10b ; 1 ≤ a < 10 |
Scientific Notation Rules
To determine the power or exponent of 10, we must follow the rule listed below:
- The base should be always 10
- The exponent must be a non-zero integer, that means it can be either positive or negative
- The absolute value of the coefficient is greater than or equal to 1 but it should be less than 10
- Coefficients can be positive or negative numbers including whole and decimal numbers
- The mantissa carries the rest of the significant digits of the number
Let us understand how many places we need to move the decimal point after the single-digit number with the help of the below representation.
- If the given number is multiples of 10 then the decimal point has to move to the left, and the power of 10 will be positive.
Example: 6000 = 6 × 103 is in scientific notation. - If the given number is smaller than 1, then the decimal point has to move to the right, so the power of 10 will be negative.
Example: 0.006 = 6 × 0.001 = 6 × 10-3 is in scientific notation.
Scientific Notation Examples
The examples of scientific notation are:
490000000 = 4.9×108
1230000000 = 1.23×109
50500000 = 5.05 x 107
0.000000097 = 9.7 x 10-8
0.0000212 = 2.12 x 10-5
Positive and Negative Exponent
When the scientific notation of any large numbers is expressed, then we use positive exponents for base 10. For example:
20000 = 2 x 104, where 4 is the positive exponent.
When the scientific notation of any small numbers is expressed, then we use negative exponents for base 10. For example:
0.0002 = 2 x 10-4, where -4 is the negative exponent.
From the above, we can say that the number greater than 1 can be written as the expression with positive exponent, whereas the numbers less than 1 with negative exponent.
Problems and Solutions
Question 1: Convert 0.00000046 into scientific notation.
Solution: Move the decimal point to the right of 0.00000046 up to 7 places.
The decimal point was moved 7 places to the right to form the number 4.6
Since the numbers are less than 10 and the decimal is moved to the right. Hence, we use a negative exponent here.
⇒ 0.00000046 = 4.6 × 10-7
This is the scientific notation.
Question 2: Convert 301000000 in scientific notation.
Solution: Move the decimal to the left 8 places so it is positioned to the right of the leftmost non zero digits 3.01000000. Remove all the zeroes and multiply the number by 10.
Now the number has become = 3.01.
Since the number is greater than 10 and the decimal is moved to left, therefore, we use here a positive exponent.
Hence, 3.01 × 108 is the scientific notation of the number.
Question 3:Convert 1.36 × 107 from scientific notation to standard notation.
Solution: Given, 1.36 × 107 in scientific notation.
Exponent = 7
Since the exponent is positive we need to move the decimal place 7 places to the right.
Therefore,
1.36 × 107 = 1.36 × 10000000 = 1,36,00,000.
Practice Questions
Problem 1: Convert the following numbers into scientific notation.
- 28100000
- 7890000000
- 0.00000542
Problem 2: Convert the following into standard form.
- 3.5 × 105
- 2.89 × 10-6
- 9.8 × 10-2
Frequently Asked Questions on Scientific Notation – FAQs
How do you write 0.00001 in scientific notation?
The scientific notation for 0.0001 is 1 × 10^{-4}.
Here,
Coefficient = 1
Base = 10
Exponent = -4
What are the 5 rules of scientific notation?
The five rules of scientific notation are given below:
1. The base should be always 10
2. The exponent must be a non-zero integer, that means it can be either positive or negative
3. The absolute value of the coefficient is greater than or equal to 1 but it should be less than 10
4. Coefficients can be positive or negative numbers including whole and decimal numbers
5. The mantissa carries the rest of the significant digits of the number
What are the 3 parts of a scientific notation?
The three main parts of a scientific notation are coefficient, base and exponent.
How do you write 75 in scientific notation?
The scientific notation of 75 is:
7.5 × 10^1 = 7.5 × 10
Here,
Coefficient = 7.5
Base = 10
Exponent = 1
How do you put scientific notation into standard form?
To convert a number from scientific notation to standard form, we should move the decimal point (if any) to the left if the exponent of 10 is negative; otherwise, proceed to the right. We must shift the decimal point as many times as the exponent indicates in power so that there will be no powers of 10 in the final representation.
How to Convert a Number to Scientific Notation
The proper format for scientific notation is a x 10^b where a is a number or decimal number such that the absolute value of a is greater than or equal to one and less than ten or, 1 ≤ |a| < 10. b is the power of 10 required so that the scientific notation is mathematically equivalent to the original number.
- Move the decimal point in your number until there is only one non-zero digit to the left of the decimal point. The resulting decimal number is a.
- Count how many places you moved the decimal point. This number is b.
- If you moved the decimal to the left b is positive.
If you moved the decimal to the right b is negative.
If you did not need to move the decimal b = 0. - Write your scientific notation number as a x 10^b and read it as “a times 10 to the power of b.”
- Remove trailing 0’s only if they were originally to the left of the decimal point.
Example: Convert 357,096 to Scientific Notation
- Move the decimal 5 places to the left to get 3.57096
- a = 3.57096
- We moved the decimal to the left so b is positive
- b = 5
- The number 357,096 converted to scientific notation is 3.57096 x 10^5
Example: Convert 0.005600 to Scientific Notation
- Move the decimal 3 places to the right and remove leading zeros to get 5.600
- a = 5.600
- We moved the decimal to the right so b is negative
- b = -3
- The number 0.005600 converted to scientific notation is 5.600 x 10^-3
- Note that we do not remove the trailing 0’s because they were originally to the right of the decimal and are therefore significant figures.
E notation is basically the same as scientific notation except that the letter e is substituted for “x 10^”.
Convert Scientific Notation to a Real Number
Multiply the decimal number by 10 raised to the power indicated.
3.456 x 10^4 = 3.456 x 10,000 = 34560
3.456 x 10^-4 = 3.456 x .0001 = 0.0003456
Example: Write 5890000 in scientific notation.
Solution:
Clearly the given number has only 3 considerable numbers/ figures since the zeroes are to be regarded as mere placeholders.
Following the first rule of scientific notation, put a decimal after the first digit in the given number.
5890000 = 5.89 × 100 × 10000
= 5.89 × 106
Similar Problems
Question 1. Write 897,000,000,000 in scientific notation.
Solution:
Clearly the given number has only 3 considerable numbers/ figures since the zeroes are to be regarded as mere placeholders.
Following the first rule of scientific notation, put a decimal after the first digit in the given number.
897,000,000,000 = 8.97 × 100 × 1000000000
= 8.97 × 102 × 109
= 8.97 × 1011
Question 2. Write 990000000000 in scientific notation.
Solution:
Clearly the given number has only 3 considerable numbers/ figures since the zeroes are to be regarded as mere placeholders.
Following the first rule of scientific notation, put a decimal after the first digit in the given number.
990000000000 = 9.9 × 10 × 10000000000
= 9.9 × 101 × 1010
= 9.9 × 1011
Question 3. Write 0.00000077 in scientific notation.
Solution:
Move the decimal point up to 7 positions to the right of 0.00000077.
To make the number 7.7, the decimal point was moved 7 places to the right.
The decimal is moved to the right since the numbers are fewer than ten. As a result, we employ a negative exponent in this case.
⇒ 0.00000077 = 7.7 × 10-7
Question 4. Write 0.0000426 in scientific notation.
Solution:
Move the decimal point up to 5 positions to the right of 0.0000426.
To make the number 4.26, the decimal point was moved 5 places to the right.
The decimal is moved to the right since the numbers are fewer than ten. As a result, we employ a negative exponent in this case.
⇒ 0.0000426 = 7.7 × 10-5
Question 5. Write 699000000 in scientific notation.
Solution:
Clearly the given number has only 3 considerable numbers/ figures since the zeroes are to be regarded as mere placeholders.
Following the first rule of scientific notation, put a decimal after the first digit in the given number.
699000000 = 6.99 × 100 × 1000000
= 6.99 × 102 × 106
⇒ 699000000 = 6.99 × 108
Question 6. Write 358000000 in scientific notation.
Solution:
Clearly the given number has only 3 considerable numbers/ figures since the zeroes are to be regarded as mere placeholders.
Following the first rule of scientific notation, put a decimal after the first digit in the given number.
358000000 = 6.99 × 100 × 1000000
= 3.58 × 102 × 106
⇒ 358000000 = 3.58 × 108
Question 7. Convert 0.00000055 into scientific notation.
Solution:
Move the decimal point up to 7 positions to the right of 0.00000055.
To make the number 5.5, the decimal point was moved 7 places to the right.
The decimal is moved to the right since the numbers are fewer than ten. As a result, we employ a negative exponent in this case.
⇒ 0.00000055 = 5.5 × 10-7
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