Scientific Notation/Examples
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Examples of Scientific Notation
Powers of 10
Various powers of $10$ are specified in scientific notation as follows:
\(\ds 10\) | \(=\) | \(\ds 10^1\) | ||||||||||||
\(\ds 100\) | \(=\) | \(\ds 10^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10 \times 10\) | ||||||||||||
\(\ds 100 \, 000\) | \(=\) | \(\ds 10^5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10 \times 10 \times 10 \times 10 \times 10\) |
Negative Powers of 10
Various powers of $10$ with negative exponent are specified in scientific notation as follows:
\(\ds 1\) | \(=\) | \(\ds 10^0\) | ||||||||||||
\(\ds 0 \cdotp 1\) | \(=\) | \(\ds 10^{-1}\) | ||||||||||||
\(\ds 0 \cdotp 01\) | \(=\) | \(\ds 10^{-2}\) | ||||||||||||
\(\ds 0 \cdotp 000 \, 01\) | \(=\) | \(\ds 10^{-5}\) |
Arbitrary Examples
Various numbers are specified in scientific notation as follows:
Example 1
\(\ds 864 \, 000 \, 000\) | \(=\) | \(\ds 8 \cdotp 64 \times 10^8\) | ||||||||||||
\(\ds 0 \cdotp 000 \, 034 \, 16\) | \(=\) | \(\ds 3 \cdotp 416 \times 10^{-5}\) |
Example 2
\(\ds 48 \, 230 \, 000\) | \(=\) | \(\ds 4 \cdotp 823 \times 10^7\) |
Example 3
\(\ds 0 \cdotp 000 \, 008 \, 4\) | \(=\) | \(\ds 8.4 \times 10^{-6}\) |
Example 4
\(\ds 0 \cdotp 000 \, 380\) | \(=\) | \(\ds 3 \cdotp 80 \times 10^{-4}\) |
Example 5
\(\ds 186 \, 000\) | \(=\) | \(\ds 1 \cdotp 86 \times 10^5\) |
Example 6
\(\ds 300 \times 10^8\) | \(=\) | \(\ds 30 \, 000 \, 000 \, 000\) |
Example 7
\(\ds 70 \, 000 \times 10^{-10}\) | \(=\) | \(\ds 0 \cdotp 000 \, 007 \, 000 \, 0\) |
Speed of Light
The speed of light is defined as:
- $c = 299 \, 792 \, 458 \text { m s}^{-1}$
In scientific notation this can be expressed as:
- $c = 2 \cdotp 99792 \, 458 \times 10^8 \text { m s}^{-1}$
Examples of Multiplication
Powers of 10
- $\paren {10^3} \paren {10^2} = 10^5$
Arbitrary Example
- $\paren {4 \, 000 \, 000} \paren {0 \cdotp 000 \, 000 \, 000 \, 2} = 0 \cdotp 000 \, 8$
Examples of Division
Powers of 10
- $\dfrac {10^6} {10^4} = 10^2$
Arbitrary Example
- $\dfrac {\paren {0 \cdotp 006} \paren {80 \, 000} } {0 \cdotp 04} = 12 \, 000$