Scientific Notation and Significant Figures

What is Scientific Notation?

It is a system that allows writing any number as the product of two factors:

Basic Structure

N \times 10^{n}

Where:

N (coefficient): A number with a single integer digit followed by decimals $1 \leq |N| < 10$
n (exponent): An integer that indicates the order of magnitude

Converting to Scientific Notation

For Large Numbers (≥ 10)

::::steps
:::step[Identify the number]
Let's take as an example: 5,700,000
:::

:::step[Move the decimal point]
Shift the decimal point to the left until we get a number between 1 and 10:

5.700000 \quad \text{(6 positions)}

:::

:::step[Determine the exponent]
The exponent is positive and equal to the number of positions moved:

5.7 \times 10^{6}

:::
::::

For Small Numbers (< 1)

::::steps
:::step[Identify the number]
Let's take as an example: 0.0065
:::

:::step[Move the decimal point]
Shift the decimal point to the right until we get a number between 1 and 10:

6.5 \quad \text{(3 positions)}

:::

:::step[Determine the exponent]
The exponent is negative and equal to the number of positions moved:

6.5 \times 10^{-3}

:::
::::

Significant Figures: What are they and why do they matter?

Practical Definition

Significant figures represent the reliable digits in a measurement or calculation. Their correct application demonstrates scientific precision in university exams.

Fundamental Rules for Identifying Them

::::steps
:::step[Rule 1: Non-zero digits]
All digits different from zero are significant.

Examples:

2.4 → 2 S.F. (2 and 4 are significant)

893 → 3 S.F. (8, 9 and 3 are significant)
:::

:::step[Rule 2: Intermediate zeros]
Zeros located between non-zero digits are significant.

Examples:

105 → 3 S.F. (1, 0 and 5 are significant)

2005 → 4 S.F. (2, 0, 0 and 5 are significant)
:::

:::step[Rule 3: Leading zeros]
Zeros to the left of the first non-zero digit are NOT significant.

Examples:

0.45 → 2 S.F. (only 4 and 5 are significant)

0.039 → 2 S.F. (only 3 and 9 are significant)
:::

:::step[Rule 4: Trailing zeros with decimal point]
Zeros to the right after the decimal point ARE significant.

Examples:

25.0 → 3 S.F. (2, 5 and 0 are significant)

3.00 → 3 S.F. (3, 0 and 0 are significant)
:::

:::step[Rule 5: Trailing zeros without decimal point]
Zeros at the end of a whole number are ambiguous.
Solution: use scientific notation to clarify.

Example for 500:

$5.00 \times 10^{2}$ → 3 S.F.

$5.0 \times 10^{2}$ → 2 S.F.

$5 \times 10^{2}$ → 1 S.F.
:::
::::

Rounding Techniques

Situation 1: Next digit less than 5

Rule

The digit and all following digits are removed without modifying the last significant figure.

Example:
35.42 → Rounded to one decimal: 35.4
Justification: 2 < 5

Situation 2: Next digit greater than 5

Rule

The last significant figure is increased by 1.

Example:
35.26 → Rounded to one decimal: 35.3
Justification: 6 > 5

Situation 3: Next digit equal to 5

Case A: There are digits after the 5

The last significant figure is increased by 1.

Example:
35.252 → Rounded to one decimal: 35.3
Justification: 5 followed by 2

Case B: The 5 is exact (no more digits)

Even-odd rule:

If the last significant figure is odd → increase by 1
If the last significant figure is even → keep it the same

Examples:
27.35 → 27.4 (3 is odd → increases)
27.25 → 27.2 (2 is even → stays the same)