Rounding and accuracy in mathematics enable simplification of numbers while retaining their essence. This section explores rounding techniques to specified degrees of accuracy, focusing on decimal places and significant figures, essential for precision and clarity in mathematical representation.

Introduction to Rounding

Rounding transforms numbers into a more manageable form, crucial across various disciplines for simplifying calculations, providing approximations, and enhancing communication.

Importance of Rounding

• Simplification: Facilitates easier calculations.
• Estimation: Offers near-accurate values swiftly.
• Clarity: Improves comprehension and dissemination of numerical data.

Rounding Techniques

Rounding to Decimal Places

Adjusting a number to a fixed number of decimal places.

• Identify the target decimal place.
• Consider the immediate next digit; increment the target if this digit is ≥5.
• Eliminate subsequent digits.

Example: $3.14159 \rightarrow 3.142$ (rounded to 3 decimal places).

Rounding to Significant Figures

Modifying a number to include a specific count of meaningful digits.

• Start at the first non-zero digit.
• Count the desired significant figures.
• Apply rounding based on the subsequent digit; increment the last counted digit if the following digit is ≥5.

Example: $0.005642 \rightarrow 0.00564$ (rounded to 3 significant figures).

Worked Examples

Example 1: Rounding to Significant Figures

Question: Round 123.456789 to 4 significant figures.

Solution:

1. Identify and Count: $123.456789 \rightarrow 1234$ (first 4 significant digits).

2. Next Digit: The 5th digit is 5.

3. Rounding: $1234 \rightarrow 1235$ because the 5th digit ≥5.

Result: $123.456789 \approx 123.5$ when rounded to 4 significant figures.

Example 2: Rounding to Decimal Places

Question: Round £23.987 to 2 decimal places.

Solution:

1. Identify Decimal Place: Targeting the 2nd decimal, $23.98$.
2. Next Digit: The 3rd decimal is 7.
3. Rounding: $23.98 \rightarrow 23.99$ since the 3rd digit ≥5.

Result: $£23.987 \approx £23.99$ rounded to 2 decimal places.