Approximation is the process of finding a value or object that is close to, but not exactly equal to, a true or precise value. The mathematical symbol for "approximately equal to" is \(\approx\).
Example:
Calculating the amount of water discharged by the Amazon river in a day.
True Value (Approximate): Volume of water is \(2,00,000\) cubic meters per second.
To find the volume discharged in a whole day:
\(24 \times 60 \times 60 = 86,400\) seconds.
Approximation/Estimation: we would multiply the approximate flow rate by the number of seconds in a day:
\(2,00,000 \times 86,400 = 17,28,00,00,000\) cubic meters in a day.
(Estimating the flow rate to \(2,00,000\), to reach the estimated daily discharge in cubic meters)
Rounding Techniques:
Rounding makes large numbers easier to manage and understand.
Rounding Up: This is used in situations where the approximated number must be more than the actual number.
Rounding Down: This is used in situations where the approximated number is intentionally less than the actual number.
Example:
1) Suppose \(732\) people are present.
Rounding Up: You would order \(750\) (or even \(800\) sweets).
To ensure every single person gets one sweet, you must round the total requirement up from \(732\) to a slightly larger, simpler number. This avoids any shortfall.
2)If the actual cost is \(₹470\).
Rounding Down: You might say the cost is around \(₹450\).
This is done to give a safe, low estimate. someone might say \(₹470\) is around \(₹470\) itself, rather than \(₹500\), to avoid overstating the cost.
Numbers can be rounded to their nearest neighbour based on a specific place value (e.g., thousand, ten thousand, lakh, crore) by using the Indian number system's place values.
Fascinating Facts about Large Numbers:
The section uses interesting facts about science, history, and nature to provide context for working with and calculating large numbers. Calculations are often presented in the form of multiplication or division problems to reveal a specific real-world quantity.
Example:
- The weight of a blue whale.
- The approximate distance between the Earth and the Sun.
- The volume of water discharged by the Amazon river.
- The total length of the longest train journey.