Approximations, Decimal Places, and Significant Figures Revision Notes for VCE SSCE General Mathematics

Approximations, Decimal Places, and Significant Figures

Introduction to approximations

Approximations help us work with numbers when exact values are impractical. Some numbers are too long to work with easily, such as $0.573\,128\,9$ or $107\,000\,000\,000$ . We round these numbers to make calculations simpler. In many situations, a specific degree of accuracy is sufficient rather than an exact answer.

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Approximations are particularly useful in real-world situations like:

Estimating costs when shopping
Reporting large statistics (population, distances in space)
Working with measurements in science and engineering
Simplifying complex calculations

Rules for rounding

When you need to round a number, follow these three simple steps:

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The Three-Step Rounding Process:

Step 1: Look at the digit immediately to the right of the position you're rounding to.

Step 2: If this digit is $5$ , $6$ , $7$ , $8$ or $9$ , round the specified digit up by one.

Step 3: If this digit is $0$ , $1$ , $2$ , $3$ or $4$ , leave the specified digit unchanged.

A helpful way to remember this: "Five or more, raise the score; four or less, let it rest."

Rounding to the nearest thousand

Let's work through an example to see how the rounding rules apply.

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Worked Example: Rounding to the Nearest Thousand

Problem: Round $34\,867$ to the nearest thousand.

Solution:

Look at the first digit to the right of the thousands place. This is $8$ .
Since $8$ is $5$ or more, we increase the thousands digit by one. The $4$ becomes a $5$ .
All digits to the right of the thousands place become zero.

$34\,867 = 35\,000$

Note that $34\,867$ is closer to $35\,000$ than to $34\,000$ , which confirms our answer makes sense.

Decimal places

Understanding decimal places is essential for working with decimal numbers accurately.

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Definition: Decimal Places

A decimal place is a position after the decimal point in a number. For example, in the number $23.798$ :

The digit $7$ is in the first decimal place
The digit $9$ is in the second decimal place
The digit $8$ is in the third decimal place

When rounding to a certain number of decimal places, you count that many positions to the right of the decimal point, then apply the rounding rules.

Rounding to decimal places

Let's examine how to round a number to a specified number of decimal places.

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Worked Example: Rounding to Two Decimal Places

Problem: Round $94.738\,295$ to two decimal places.

Solution:

For two decimal places, count two positions to the right of the decimal point. This takes us to the digit $3$ .
Look at the digit to the right of this position, which is $8$ .
Since $8$ is " $5$ or more", we increase the digit in the second decimal place by one. The $3$ becomes $4$ .
Write your answer.

$94.738\,295 = 94.74 \text{ (to two decimal places)}$

Scientific notation (standard form)

Scientific notation, also called standard form, is a way to express very large or very small numbers efficiently. This notation is particularly useful in science and mathematics.

To write a number in scientific notation, you express it as a number between 1 and 10 (including $1$ but not $10$ ), multiplied by a power of 10.

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Examples of Scientific Notation:

$134.7$ written in scientific notation is $1.347 \times 10^{2}$
$0.0823$ written in scientific notation is $8.23 \times 10^{-2}$

Writing numbers in scientific notation

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Worked Example: Converting to Scientific Notation

Problem: Write the following numbers in scientific notation: a) $7\,800\,000$
b) $0.000\,000\,5$

Solution:

Part a:

Place a decimal point to the right of the first non-zero digit: $7.800\,000$
Count how many places the decimal point needs to move. Starting from $7.8$ , we need to move $6$ places to the right to reach $7\,800\,000$ .
To move the decimal point $6$ places to the right, we multiply by $10^{6}$ .

$7\,800\,000 = 7.8 \times 10^{6}$

Part b:

Place a decimal point after the first non-zero digit: $5.0$
Count how many places the decimal point needs to move. Starting from $5.0$ , we need to move $7$ places to the left to reach $0.000\,000\,5$ .
To move the decimal point $7$ places to the left, we multiply by $10^{-7}$ .

$0.000\,000\,5 = 5.0 \times 10^{-7}$

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Understanding Powers of 10:

A positive power of 10 indicates a large number (decimal point moves right), while a negative power of 10 indicates a small number (decimal point moves left).

Converting from scientific notation to basic numerals

Sometimes you need to convert a number from scientific notation back to its standard decimal form.

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Worked Example: Converting from Scientific Notation

Problem: Write the following scientific notation numbers as basic numerals: a) $3.576 \times 10^{7}$
b) $7.9 \times 10^{-5}$

Solution:

Part a:

Multiplying by $10^{7}$ means moving the decimal point $7$ places to the right.
Starting from $3.576$ , move the decimal point $7$ places right. Add zeros as needed.

$3.576 \times 10^{7} = 35\,760\,000$

Part b:

Multiplying by $10^{-5}$ means moving the decimal point $5$ places to the left.
Starting from $7.9$ , move the decimal point $5$ places left. Add zeros as needed.

$7.9 \times 10^{-5} = 0.000\,079$

Significant figures

Significant figures are the digits in a number that indicate its accuracy. They tell us how precise a measurement or calculation is.

For example, when a football match attendance is reported as $64\,000$ people, this means the crowd was estimated to the nearest thousand. The digits $6$ and $4$ are significant because they make a claim about accuracy. We say $64\,000$ is accurate to two significant figures.

In contrast, $64\,208$ includes five significant figures: $6$ , $4$ , $2$ , $0$ and $8$ . This number claims accuracy to the individual person.

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Key Principle:

All digits that appear in scientific notation are regarded as significant figures.

This makes scientific notation particularly useful for clearly showing the precision of a number.

Understanding significant figures with scientific notation

When working with numbers that have many zeros, such as $26\,508\,000\,000$ or $0.000\,000\,076$ , the zeros can cause confusion. The best approach is to first write the number in scientific notation, then round to the required number of significant figures.

For example, starting with $26\,508\,000\,000$ :

\begin{aligned} 26\,508\,000\,000 &= 2.6508 \times 10^{10} \text{ (5 significant figures)} \\ &= 2.651 \times 10^{10} \text{ (4 significant figures)} \\ &= 2.65 \times 10^{10} \text{ (3 significant figures)} \\ &= 2.7 \times 10^{10} \text{ (2 significant figures)} \\ &= 3 \times 10^{10} \text{ (1 significant figure)} \end{aligned}

Similarly, for $0.000\,000\,076$ :

\begin{aligned} 0.000\,000\,076 &= 7.6 \times 10^{-8} \text{ (2 significant figures)} \\ &= 8 \times 10^{-8} \text{ (1 significant figure)} \end{aligned}

Rounding to significant figures

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Worked Example: Rounding to Significant Figures

Problem: Write each number in scientific notation, then round to two significant figures: a) $9764.809\,4$
b) $0.000\,004\,716\,8$

Solution:

Part a:

Write in scientific notation by placing the decimal point after the first non-zero digit and multiplying by the appropriate power of $10$ .

$9764.809\,4 = 9.764\,809\,4 \times 10^{3}$

To round to two significant figures, look at the third digit. Since $6$ is greater than $5$ , round the second digit up from $7$ to $8$ .

$9.8 \times 10^{3} = 9\,800$

Part b:

Write in scientific notation.

$0.000\,004\,716\,8 = 4.716\,8 \times 10^{-6}$

To round to two significant figures, look at the third digit. Since $1$ is not greater than or equal to $5$ , leave the second digit unchanged.

$4.7 \times 10^{-6} = 0.000\,004\,7$

Summary

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Key Points to Remember:

Rounding rule: Look at the digit to the right of where you're rounding. If it's $5$ or more, round up. If it's $4$ or less, leave unchanged.
Decimal places: Count positions after the decimal point. The first decimal place is the first digit after the point.
Scientific notation: Express numbers as a value between $1$ and $10$ multiplied by a power of $10$ . Positive powers for large numbers, negative powers for small numbers.
Significant figures: Show the accuracy of a number. Convert to scientific notation first, then round to the required number of significant figures.
Converting from scientific notation: Move the decimal point right for positive powers of $10$ , left for negative powers, adding zeros as needed.

Approximations, Decimal Places, and Significant Figures (VCE SSCE General Mathematics): Revision Notes