2.1 Significant Figures
Significant figures (s.f.) refer to the digits in a number that are determined accurately to a given degree of precision.
Rules for Determining Significant Figures
- Non-zero digits: All non-zero digits are always significant. (e.g., 547 has 3 s.f.)
- Zeros between non-zero digits: Always significant. (e.g., 5007 has 4 s.f., 3.02 has 3 s.f.)
- Leading zeros: Zeros before the first non-zero digit in decimals less than 1 are NOT significant. They serve only as place holders. (e.g., 0.0034 has 2 s.f.)
- Trailing zeros in decimals: Zeros at the end of a decimal number are significant. (e.g., 4.200 has 4 s.f., 0.050 has 2 s.f.)
- Trailing zeros in whole numbers: Zeros at the end of a whole number may or may not be significant depending on the degree of accuracy required. (e.g., 8 000 rounded to the nearest thousand has 1 s.f., to the nearest hundred has 2 s.f., to the nearest ten has 3 s.f.)
Rounding Numbers to Specific Significant Figures
- Count the required number of significant figures from left to right starting with the first non-zero digit.
- Look at the digit immediately after the target position:
- If it is 5 or greater, add 1 to the last target digit (round up).
- If it is less than 5, keep the last target digit unchanged (round down).
- Replace remaining digits before the decimal point with zeros if necessary.
Example: Round $0.04567$ to $2$ s.f. $\rightarrow$ Target digit is $5$, next digit is $6 \ge 5$, so round up to $0.046$.
2.2 Standard Form
Standard Form (scientific notation) is a way to express very large or very small numbers in the form:
$$A \times 10^n \quad \text{where } 1 \le A < 10 \text{ and } n \text{ is an integer}$$
Converting Numbers to Standard Form
- Numbers $\ge 10$: $n$ is a positive integer representing how many places the decimal point moves to the left.
Example: $456 \text{ } 000 = 4.56 \times 10^5$
- Numbers $< 1$: $n$ is a negative integer representing how many places the decimal point moves to the right.
Example: $0.00078 = 7.8 \times 10^{-4}$
2.3 Basic Operations Involving Numbers in Standard Form
1. Addition and Subtraction
To add or subtract, first ensure that the powers of $10$ are the same. Factor out $10^n$ before performing the operation:
$$A \times 10^n + B \times 10^n = (A + B) \times 10^n$$
$$A \times 10^n - B \times 10^n = (A - B) \times 10^n$$
Example: $3.2 \times 10^5 + 4.5 \times 10^4 = 3.2 \times 10^5 + 0.45 \times 10^5 = (3.2 + 0.45) \times 10^5 = 3.65 \times 10^5$
2. Multiplication
Multiply the single-digit numbers and add the exponents using the multiplication law of indices:
$$(A \times 10^m) \times (B \times 10^n) = (A \times B) \times 10^{m+n}$$
Example: $(2.5 \times 10^4) \times (3 \times 10^3) = (2.5 \times 3) \times 10^{4+3} = 7.5 \times 10^7$
3. Division
Divide the single-digit numbers and subtract the exponents using the division law of indices:
$$(A \times 10^m) \div (B \times 10^n) = (A \div B) \times 10^{m-n}$$
Example: $(8 \times 10^6) \div (2 \times 10^{-3}) = (8 \div 2) \times 10^{6 - (-3)} = 4 \times 10^9$
4. Normalising Results
If the calculated value of $A$ falls outside the range $1 \le A < 10$, adjust $A$ and update $n$ accordingly:
Example: $12 \times 10^4 = (1.2 \times 10^1) \times 10^4 = 1.2 \times 10^5$