Topics
1 The real numbers
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Australia
Year 9

1.03 Introducing scientific notation

Lesson
Worksheet
Practice
Lesson

Scientific notation

Scientific notation or standard form is a compact way of writing very big or very small numbers. As the name suggests, scientific notation is frequently used in science. For example:

  • The sun has a mass of approximately 1.988\times 10^{30} kg which is much easier to write than 1988\,000\,000\,000\,000\,000\,000\,000\,000\,000 kg.

  • The mass of an atom of Uranium (one of the heaviest elements) is only approximately 3.95\times 10^{-22} g. That is 0.000\,000\,000\,000\,000\,000\,000\,395 g.

Exploration

The following applet demonstrates how a decimal number is written in the scientific notation. Try entering various numbers up to 10 decimal places.

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In writing a scientific notation, the coefficient or decimal number must be greater than or equal to 1 and less than 10, and the power of ten can be determined by counting how many places the decimal point has shifted.

In scientific notation, numbers are written in the form a\times 10^n, where a is a decimal number between 1 and 10 and n is an integer (positive or negative).

  • A negative index indicates how many factors of ten smaller than a the value is.

  • A positive index indicates how many factors of ten larger than a the value is.

  • A index of zero indicates that the value is a because 10^0=1.

Examples

Example 1

Express 0.000\,347 in scientific notation.

Worked Solution
Create a strategy

Use the scientific notation form a \times 10^{n}, where a is a decimal number between 1 and 10 and n is a negative integer.

Apply the idea

To find the first part of our scientific notation we place the decimal point after the first non-zero number, soa=3.47

0.000\,347 is 10\,000 or 10^{4} times smaller than 3.47. So:0.000\,347=3.47 \times 10^{-4}

Example 2

Express the following number as a basic numeral: 2\times 10^7

Worked Solution
Create a strategy

Use the fact that 10^7=10\,000\,000.

Apply the idea
\displaystyle 2\times 10^7\displaystyle =\displaystyle 2\times 10\,000\,000Evaluate 10^7
\displaystyle =\displaystyle 20\,000\,000Evaluate the multiplication

Example 3

Express the following number in scientific notation: 84\,245\,000

Worked Solution
Create a strategy

Use the scientific notation form a \times 10^{n}, where a is a decimal number between 1 and 10 and n is a positive integer.

Apply the idea

To find the first part of our scientific notation we place the decimal point after the first non-zero number, so:a=8.4245

84\,245\,000 is 10\,000\,000 or 10^{7} times bigger than 8.4245. So:84\,245\,000=8.4245 \times 10^{7}

Example 4

Express the following number as a decimal number: 3.62\times 10^{-4}

Worked Solution
Create a strategy

Decrease the place value of each digit by 4 places.

Apply the idea
\displaystyle 3.62\times 10^{-4}\displaystyle =\displaystyle 0.000\,362Move the decimal point left 4 places
Idea summary

In scientific notation, numbers are written in the form a\times 10^n, where a is a decimal number between 1 and 10 and n is an integer (positive or negative).

  • A negative index indicates how many factors of ten smaller than a the value is.

  • A positive index indicates how many factors of ten larger than a the value is.

  • A index of zero indicates that the value is a because 10^0=1.

Outcomes

ACMNA209

Apply index laws to numerical expressions with integer indices

ACMNA210

Express numbers in scientific notation

ACMNA212

Extend and apply the index laws to variables, using positive integer indices and the zero index

ACMMG219

Investigate very small and very large time scales and intervals

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