Mathematics- Rounding

General Rules of Rounding

Rounding simplifies numbers while keeping their value close to the original. The basic procedure depends on the digit immediately to the right of the target place value:

• Identify the target place value (e.g., nearest ten, hundredth, or significant figure).

• Look at the decision digit directly to its right:

  • Round Up: If the decision digit is $5$ or greater ($5, 6, 7, 8, 9$), increase the target digit by $1$ and change all following digits to zero (or drop them if they are after a decimal point).

  • Round Down: If the decision digit is less than $5$ ($0, 1, 2, 3, 4$), keep the target digit the same and change all following digits to zero (or drop them if they are after a decimal point).

Rounding Whole Numbers

When rounding to a specific whole number place value:

1. Rounding to the Nearest Ten

   • Look at the units (ones) digit.

   • Example: Rounding $84$ to the nearest ten gives $80$.

   • Example: Rounding $87$ to the nearest ten gives $90$.

2. Rounding to the Nearest Hundred

   • Look at the tens digit.

   • Example: Rounding $452$ to the nearest hundred gives $500$.

Rounding Decimals

Decimals are often rounded to a specified number of decimal places (d.p.):

• 1 Decimal Place (Tenths): Look at the hundredths digit.

  • Example: $3.14159 \rightarrow 3.1$

• 2 Decimal Places (Hundredths): Look at the thousandths digit.

  • Example: $3.14159 \rightarrow 3.14$

• 3 Decimal Places (Thousandths): Look at the ten-thousandths digit.

  • Example: $3.14159 \rightarrow 3.142$

Rounding to Significant Figures (s.f.)

Significant figures reflect the precision of a measurement, counted from left to right starting at the first non-zero digit.

1. Identifying Significant Figures

   • Non-zero digits are always significant (e.g., $42.3$ has $3$ s.f.).

   • Zeros between non-zero digits are significant (e.g., $506$ has $3$ s.f.).

   • Leading zeros are not significant (e.g., $0.0025$ has $2$ s.f.).

   • Trailing zeros in a decimal number are significant (e.g., $4.300$ has $4$ s.f.).

2. Rounding Process

   • Count to the required number of significant figures from the first non-zero digit.

   • Check the next digit to determine whether to round up or down.

   • Example: Round $0.004738$ to $2$ significant figures.

     • First non-zero digit is $4$.

     • Second significant figure is $7$.

     • The following digit is $3$ (less than $5$), so round down to $0.0047$.