Scientific Notation and Logarithmic Scales

In short: Scientific notation writes a number as $a\times 10^n$ with $1\le a<10$, which makes very large and very small numbers easy to handle. On a logarithmic scale, like the Richter scale, decibels or pH, each step up means 10 times bigger.

Starting point: Years 9 and 10

Scientific notation

A number in scientific notation is written as

$$a\times 10^n\text{where }1\le a<10\text{ and }n\text{ is a whole number}$$

• Large numbers have a positive power: $45000=4.5\times 10^4$.
• Small numbers have a negative power: $0.0032=3.2\times 10^{-3}$.

Count how many places the decimal point moves. Moving left makes the power go up, moving right makes it go down.

Significant figures

Significant figures are the digits that carry meaning. Count from the first non-zero digit.

• 3.047 has 4 significant figures.
• 0.0052 has 2 (the leading zeros only hold the place).
• $6.10\times 10^5$ has 3 (the final zero was measured).

To round 186 420 to 2 significant figures: 190 000, or $1.9\times 10^5$.

Logarithmic scales

On a normal scale, equal steps add the same amount. On a logarithmic scale, equal steps multiply by the same amount, usually 10.

$${\log }_{10}10=1{\log }_{10}100=2{\log }_{10}1000=3$$

The log tells you the power of 10, so a log scale is really a scale of powers.

Quick summary

• Scientific notation: $a\times 10^n$ with $1\le a<10$.
• Multiply: multiply the numbers and add the powers. Divide: divide and subtract the powers.
• Significant figures start at the first non-zero digit.
• On a log scale each step multiplies by 10. A difference of $d$ steps means $10^d$ times.
• Richter, decibels and pH are log scales.

Quiz