Instructions for use:
- If the scientific notation is 0.1244107 x 10^{3}, you must use 0.1244107e+3;
- If the scientific notation is 14478 x 10^{-3}, you must use 14478e-3;
- You will replace base 10 with the letter (e).
Scientific Notation | Decimal | |
Million in scientific notation | 1 × 10^{6} | 1,000,000 |
10 Million in scientific notation | 1 × 10^{7} | 10,000,000 |
100 Million in scientific notation | 1 × 10^{8} | 100,000,000 |
Billion in scientific notation | 1 × 10^{9} | 1,000,000,000 |
10 Billion in scientific notation | 1 × 10^{10} | 10,000,000,000 |
100 Billion in scientific notation | 1 × 10^{11} | 100,000,000,000 |
Trillion in scientific notation | 1 × 10^{12} | 1,000,000,000,000 |
10 Trillion in scientific notation | 1 × 10^{13} | 10,000,000,000,000 |
100 Trillion in scientific notation | 1 × 10^{14} | 100,000,000,000,000 |
Scientific Notation |
Decimal | |
Giga in scientific notation |
10^{9} |
1,000,000,000 |
Mega in scientific notation |
10^{6} |
1,000,000 |
Kilo in scientific notation |
10^{3} |
1,000 |
Hecto in scientific notation |
10^{2} |
100 |
Deka in scientific notation |
10^{1} |
10 |
Deci in scientific notation |
10^{-1} |
0.1 |
Centi in scientific notation |
10^{-2} |
0.01 |
Milli in scientific notation |
10^{-3} |
0.001 |
Micro in scientific notation |
10^{-6} |
0.000001 |
Nano in scientific notation |
10^{-9} |
0.000000001 |
Pico in scientific notation |
10^{-12} |
0.000000000001 |
Scientific Notation: Addition and Subtraction
Scientific Notation: Multiplication and Division
In every field of science, one must have to deal with numbers, the numbers can be in any format like the weight of the earth in kilograms is a very large number or mass of an electron is a very small number. The representation of these types of numbers is not easy.
Let suppose there is a number that has 22 zeros and written in this way
60220000000000000000000000
It’s not easy to read or understand that number easily. So there is a way known as a scientific notation that we can use to write very large or small numbers in an easier and understandable way.
Importance of scientific notation
In 1998 NASA launched the orbiter to find the data of climate change at mars but after three years the orbiter got lost and investigation on this matter shows that estimate of data was wrong because of two teams transfer their data in different units. The representation of the number in the standard form is very important because the accuracy of numbers matters, while reading too large or too small number we can make mistake while counting 0’s.
How to express numbers in scientific notation?
The numbers are written in product from like the first number is mantissa and second is the power of 10 exponents:
Any number = mantissa x 10^{exponent}
For example, if we take any large number of their scientific a notation will be like this way
7,000,000,000,000,000,000,000 = 7 × 10^{21}
Here 7 is mantissa and 21 is exponent. Now we can see that while reading and writing of these types of numbers the chances of error become reduced and we can easily use these numbers in any paper writing work.
Before using scientific notation let’s find underlying theory. How we can find the different powers for 10?
- 10^{0} = 1
- 10¹ = 10
- 10³ = 1000
Here 10 power helps to find the zeros, like when power is 0 there is no zero follow 1, when power is 3 there are 3 zeros follows 1. Like in a similar way 10^{100} = 1000….0 represent 100 zeros. Which is a very large number and is not easy to read so by scientific notation the representation of very big number become easy.
To express numbers in scientific notation, there should be only one digit remains to the left of the decimal point for this the decimal point move towards the left or right-hand side depending upon the number, either it is greater or less than zero. Multiplying by the different exponent of allow us to move the decimal place.
For example in
9.2867 × 10^{4} = 92876.00
the exponent of 10 is positive it allows decimal to move towards the right. While on the other hand, negative exponent move the decimal towards left
5 × 10^{-15} = 0.000000000000005
But we keep one the thing in mind in the standard form of scientific notation that we have to left just one digit to the left of the decimal.
More advantage of scientific notation is that we can easily add, subtract, multiply, and divide large numbers. Let’s see how we can use this method to add numbers.
To add or subtract numbers in scientific notation the following steps are uses:
- the exponents of 10 should keep the same in both terms.
- After making the exponents of 10 the same, we can add or subtract the numbers.
- Lastly we should adjust the answer according to scientific notation.
Add Scientific Notation: (5.7 × 10^{4}) + (2.25 × 10^{5})
- Adjust the power of large exponent according to the small power.
5.7 × 10^{4} ) + (2.25 × 10^{1} × 10^{4})
- Now group the numbers
(5.7 + 2.25 × 10^{1}) × 10^{4}
(5.7 + 22.5 ) × 10^{4}
28.2 × 10^{4}
- Now adjust number according to standard form of scientific notation
(2.82 × 10^{1}) × 10^{4} = 2.82 × 10^{5}
In the similar way we can subtract two quantities.
Subtract Scientific Notation: (6.67 × 10^{8} ) – (8.4 × 10^{6})
- Adjust the power of large exponent according to the small power.
(6.67 × 10^{2} × 10^{6}) – (8.4 × 10^{6})
- Now group the numbers
(667 – 8.4) × 10^{6}
658.6 × 10^{6}
- Now adjust number according to standard form of scientific notation
(6.58 × 10^{2}) × 10^{6} = 6.58 × 10^{8}
All numbers in scientific notation have the base 10 so we can easily multiply and divide them:
- Multiply and divide two numbers, we multiply or divide their mantissas, add and subtract the power of exponents respectively.
- In both cases, we should convert a number in the standard form of scientific notation.
For example:
Multiply Scientific Notation: (6.87 × 10^{12}) × (4.102 × 10^{6})
- Multiply the mantissas of each quantity
6.87 × 4.102 = 28.187
- Now add the powers of exponents
(10^{12} × 10^{6}) = 10^{18}
- Write the number in standard form of scientific notation
28.187 × 10^{18} = 2.8187 × 10^{1} × 10^{18} = 2.8187 × 10^{19}
Let’s try example for division:
Divide Scientific Notation: (2.04 × 10^{6}) ÷ (7.82 × 10^{2})
- Divide the mantissas of each quantity
2.04 ÷ 7.82 = 0.26086
- Now subtract the powers of exponents
(10^{6} × 10²) = 10^{4}
- Write the number in standard form of scientific notation
0.26086 × 10^{4} = 2.6086× 10^{-1} × 10^{4} = 2.6086 × 10^{3}