Question

Round each of the following numbers to three significant figures: (a) \(8.8726\) (b) \(21.25\) (c) \(129.509\) (d) \(1.995 \times 10^{6}\)

Short answer

8.87, 21.3, 130, 2.00 × 10^6

Step-by-step solution

Identify the First Three Significant Figures

For each number, identify and underline the first three significant figures. These are the figures that will remain after rounding.

Check the Fourth Figure

Examine the fourth figure to decide whether to round up or keep the third figure the same. If the fourth figure is 5 or above, round the third figure up. If it's less than 5, keep the third figure the same.

Apply Rounding

Perform the rounding based on the fourth figure, then adjust the number if necessary to maintain three significant figures.

Write the Numbers in Scientific Notation (if needed)

For very large or very small numbers, write the result in scientific notation.

Key concepts

Number Rounding

Number rounding is a way to reduce a number to a more manageable or simpler form. When rounding, you select which decimal place or whole digit to round to and adjust the number accordingly.
Let’s break this down: If you're rounding to three significant figures, like in our exercise, you first identify the first three significant figures. Then check the digit immediately after these three figures.

• If this fourth figure is 5 or higher, you round the third figure up by one.
• If this fourth figure is less than 5, you leave the third figure as it is and drop all subsequent digits.

For example, rounding the number 8.8726 to three significant figures:
- Identify the first three figures: 8.87
- Check the fourth figure: 2 (which is less than 5)
- So, the rounded number to three significant figures is 8.87.
This method helps to maintain precision without retaining unnecessary details.

Scientific Notation

Scientific notation is a way to express very large or very small numbers succinctly. It converts numbers into the form of a coefficient between 1 and 10 multiplied by a power of 10.
For example, the number 1,995,000 can be written as: 1.995 x 10^6.
To write a number in scientific notation:

• Move the decimal point to position it after the first non-zero digit.
• Count how many places you moved the decimal point; this becomes the exponent of 10.
• If you moved the decimal to the left, the exponent is positive; if to the right, it's negative.

Take for instance, 0.00003 written in scientific notation becomes 3 x 10^-5, because the decimal is moved 5 places to the right.
This notation is particularly useful for dealing with extremely large or tiny numbers in calculations.

Significant Digits in Mathematics

Significant digits (or significant figures) are crucial in mathematics as they reflect the precision of a number. They include all the digits that contribute to the number's accuracy.
Consider these rules for identifying significant digits:

• All non-zero numbers are always significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant; they only hold the decimal point.
• Trailing zeros in a decimal number are significant.

For instance, in the example 129.509, all the digits are significant as they provide precision to the number. When you round this to three significant figures:
- Identify the first three figures: 129
- Check the fourth figure: 5
- Since the fourth figure is 5 or above, round up the third figure.
- The rounded number is 130.
Understanding significant digits ensures precision and accuracy in mathematical calculations and scientific measurements.