Standard Form and Significant Figures Revision Notes for HSC SSCE Mathematics Standard

Standard Form and Significant Figures

Understanding standard form

When working with extremely large or tiny numbers, standard form (also known as scientific notation) provides a more manageable way to express them. This method involves taking a number that falls between 1 and 10, then multiplying it by 10 raised to a certain power.

For instance, the number 4,100,000 can be written in standard form as $4.1 \times 10^6$ . The exponent (power of 10) tells us how many times we multiply 10 by itself. We can verify this:

$4.1 \times 10^6 = 4.1 \times (10 \times 10 \times 10 \times 10 \times 10 \times 10) = 4,100,000$

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Key Rule for Powers of 10:

A helpful rule to remember is that large numbers require a positive power of 10, whilst small numbers need a negative power of 10. This makes sense because large numbers involve moving the decimal point to the right (making them bigger), whilst small numbers involve moving it to the left (making them smaller).

Writing numbers in standard form

To convert any number into standard form, follow this systematic approach:

Step 1: Locate the first two non-zero digits in your number.

Step 2: Place a decimal point between these two digits. This creates your number between 1 and 10.

Step 3: Count how many places the decimal point has moved from its original position to its new position. This count becomes your power of 10.

Step 4: Determine whether the power should be positive or negative. Use a positive power for numbers greater than 10, and a negative power for numbers less than 1.

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Worked Example: Expressing Earth's Land Surface Area

Let's express 153,400,000 square kilometres (the approximate land surface of Earth) in standard form.

Step 1: The first two non-zero digits are 1 and 5.

Step 2: Place the decimal point between them: $1.534$

Step 3: Count the digits from the original decimal point (at the end of 153,400,000) to the new position: 8 digits

Step 4: Since this is a large number, the power is positive: $+8$

Step 5: Write the final answer: $153,400,000 = 1.534 \times 10^8$

Understanding significant figures

Significant figures tell us how precise a number is by identifying which digits actually carry meaningful information. They include all digits that contribute to the number's accuracy, but exclude certain zeros that serve merely as placeholders.

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Rules for Counting Significant Figures:

All non-zero digits are always significant
Zeros between non-zero digits are significant
Zeros after a decimal point and after a non-zero digit are significant
Leading zeros (zeros at the start of a number) are NOT significant
Trailing zeros in whole numbers (without a decimal point) are NOT significant

Let's look at some examples to clarify these rules:

$51.340$ has five significant figures: the digits 5, 1, 3, 4, and the final 0 all count
$0.00871$ has three significant figures: only 8, 7, and 1 count (the leading zeros don't)
$56091$ has five significant figures: all digits count, including the zero in the middle

chatImportant

Sometimes the number of significant figures in a whole number without a decimal point can be ambiguous. For example, does 8000 have one, two, three, or four significant figures? To clarify this, the last significant digit can be underlined. For instance, $80\underline{0}0$ clearly indicates two significant figures.

Rounding to significant figures

When you need to round a number to a specific number of significant figures, standard form makes the process straightforward:

Step 1: Convert the number into standard form.

Step 2: Count the digits in the coefficient (the number between 1 and 10) to determine the current accuracy. Remember to ignore trailing zeros at the end of whole numbers.

Step 3: Round the coefficient to the required number of significant figures, then write your answer in standard form.

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Worked Example: Rounding to Significant Figures

Let's round these numbers to the specified number of significant figures:

Part a: Write 153,400,000 correct to three significant figures

Step 1: Write in standard form:

$153,400,000 = 1.534 \times 10^8$

Step 2: Count the digits:

$1.534$ has 4 digits

Step 3: Round to three significant figures:

$1.534$ rounds to $1.53$

Step 4: Write the final answer:

$153,400,000 = 1.53 \times 10^8 \text{ (to 3 sig. fig.)}$

Part b: Write 0.000657 correct to two significant figures

Step 5: Write in standard form:

$0.000657 = 6.57 \times 10^{-4}$

Step 6: Count the digits:

$6.57$ has 3 digits

Step 7: Round to two significant figures:

$6.57$ rounds to $6.6$

Step 8: Write the final answer:

$0.000657 = 6.6 \times 10^{-4} \text{ (to 2 sig. fig.)}$

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Key Points to Remember:

Standard form expresses numbers as a value between 1 and 10 multiplied by a power of 10, written as $a \times 10^n$ where $1 \leq a < 10$
Large numbers use positive powers of 10, whilst small numbers use negative powers of 10
Significant figures indicate the precision of a number by counting meaningful digits
Leading zeros and trailing zeros in whole numbers (without decimal points) are NOT significant figures
To round to a specific number of significant figures, convert to standard form first, then round the coefficient and rewrite in standard form

Standard Form and Significant Figures (HSC SSCE Mathematics Standard): Revision Notes