- Expressing Numbers in Standard Form
Standard form (or scientific notation) is a method for writing very large or small numbers. It consists of two parts:
- A number between 1 and 10.
- A power of 10.
- Expressing Large Numbers
Move the decimal point from left to right; the power of 10 will be positive.
- Examples:
- 4500 = 4.5 × 10³
- 67,413 = 6.7413 × 10⁴
- 300,000,000 = 3.0 × 10⁸
- Expressing Small Numbers
Move the decimal point from right to left; the power of 10 will be negative.
- Examples:
- 0.00067 = 6.7 × 10⁻⁴
- 0.00145 = 1.45 × 10⁻³
- 0.335 = 3.35 × 10⁻¹
- Significant Figures
Significant figures are digits that reflect the precision of a number.
Guidelines:
- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
- Leading zeros (before non-zero digits) are not significant.
- Trailing zeros after a decimal point are significant.
- In whole numbers without a decimal point, the least significant figure is the rightmost non-zero digit.
Examples:
- 6753 has 4 significant figures.
- 40072 has 5 significant figures.
- 0.0089 has 2 significant figures.
- 9.0 has 2 significant figures.
Rounding:
- Round up if the next digit is 5 or more; round down if 4 or less.
- 14.628 → 14.63 (4 significant figures)
- 15.473 → 15.47 (4 significant figures)
- Expressing Results with Significant Figures
When performing operations, the result should reflect the least number of significant figures from the inputs.
Examples:
- Addition/Subtraction:
2345 + 7800 + 934,456 = 940,000 (2 significant figures). - Multiplication/Division:
(2.467 × 465) ÷ 2.7 = 420 (2 significant figures).
- Accuracy vs. Precision
- Accuracy: How close a measurement is to the actual value.
- Precision: How close multiple measurements are to each other.
Example:
If three measurements are 30.01g, 30.02g, and 30.03g, they are both accurate (close to actual value 30.0g) and precise (close to each other).