Activity 1
The Problem with Very Large and Very Small Numbers
Earth
Consider the following two measurements:
Mass of the Earth:5,972,000,000,000,000,000,000,000 kg
Mass of an electron:0.000000000000000000000000000009109 kg
Observation
- Writing such a number once is manageable, but what if we have to copy it again and again?
- If even a single zero is accidentally written one more or one less while copying, the value changes completely.
Conclusion
- There is a serious risk of error while writing very large or very small numbers.
- Therefore, we need a shorthand system to represent such numbers accurately and efficiently.
Activity 2
Converting a Large Number
Let us consider the number 3500 and shift its decimal point step by step.
Step-by-Step Working
$3500 = 3500.0$
decimal after units digit
$= 350.00 \times 10$
$= 350.00 \times 10^1$
decimal moves one digit left
$= 35.000 \times 10 \times 10$
$= 35.000 \times 10^2$
decimal moves one digit left
$= 3.5000 \times 10^2 \times 10$
$= 3.5000 \times 10^3$
decimal moves one digit left
Observation: When the decimal point moves from right to left, the power of 10 increases.
Activity 3
Converting a Small Number
Now let us consider the number 0.00037 and shift its decimal point step by step.
Step-by-Step Working
$0.00037 = 0.00037 \times 10^0$
decimal after units digit
$= 00.0037 \times \dfrac{10^0}{10}$
$= 00.0037 \times 10^{-1}$
decimal moves one digit right
$= 000.037 \times \dfrac{10^{-1}}{10}$
$= 000.037 \times 10^{-2}$
decimal moves one digit right
$= 0000.37 \times \dfrac{10^{-2}}{10}$
$= 0000.37 \times 10^{-3}$
decimal moves one digit right
$= 00003.7 \times \dfrac{10^{-3}}{10}$
$= 00003.7 \times 10^{-4}$
$= 3.7 \times 10^{-4}$
decimal moves one digit right
Observation: When the decimal point moves from left to right, the power of 10 decreases.
Conclusion & Definition
What is Scientific Notation?
Conclusion
- Both large and small numbers are expressed as a power of 10.
- There is only one digit before the decimal point.
- This method of writing very big or very small numbers is called Scientific Notation.
Definition
The way of expressing very large or very small numbers as a power of 10, with only one digit before the decimal point, is called Scientific Notation.
General Form
$$a \times 10^n \quad \text{where } 1 \leq a < 10 \text{ and } n \in \mathbb{Z}$$