Scientific Notation & Significant Figures - Quick Review

Scientific notation uses a coefficient between 1 and less than 10 multiplied by a power of 10: c \times 10^{n}, \quad 1 \le c < 10, \quad n \in \mathbb{Z}
Examples: $5.7 \times 10^{26}$ kg, $2.0 \times 10^{-7}$ m
Positive exponent means a large number; negative exponent means a small number (division by powers of 10)
- Example: $1.73 \times 10^{3} = 1730$
- Example: $1.0 \times 10^{-3} = 0.001$
Expanded (standard) notation is the non-scientific form; both represent the same value
To move from standard to scientific notation: adjust the decimal so the coefficient is in [1, 10) and apply the proper power of 10
To convert back, move the decimal point left for positive exponents and right for negative exponents

Coefficient must satisfy: 1 \le c < 10
The coefficient digits are significant; leading zeros are not allowed in the coefficient
In scientific notation, all digits in the coefficient are significant

Leading zeros (zeros to the left of the first nonzero digit) are not significant: $0.0035\quad\text{has two sig figs (3 and 5)}$
Zeros between nonzero digits are significant ("zero sandwiches"); e.g., $908\,\text{has 3 sig figs}$ ; $1\,0\,0\,5\quad\text{has 3 or more depending on decimal point}$
Zeros to the right of a decimal point are significant
Zeros at the end of a number without a decimal point are not significant; with a decimal point, they are
In a coefficient of scientific notation, zeros count toward significance as they are part of the coefficient

Nonzero digits are always significant
Zeros between nonzero digits are significant
Zeros to the right of a decimal point are significant
Zeros before the first nonzero digit are not significant (leading zeros)
Zeros at the end of a decimal number are significant (e.g., 12.300 has four sig figs)
Zeros at the end of a number without a decimal are not significant (e.g., 1200 has 2 sig figs unless clarified by notation)
In scientific notation, the coefficient conveys all significant digits; e.g., $4.520 \times 10^{5}$ has 4 sig figs

A measurement is as precise as the instrument allows; precision dictates sig figs
Example with a ruler: 1.75 cm is a reasonable estimate (three sig figs) if the instrument supports that precision
A measurement may be written with an estimated digit beyond the smallest marked division
Different scales on the same instrument have different precisions (e.g., a fine scale vs a coarse scale) and affect the reported value

If a value is given as 4.08 cm (with a coarse scale only up to 0.01 cm), reporting 4.08 implies the last digit (the 8) is significant; if not supported by the instrument, you would round to the least precise place (e.g., 4.1 cm or 4.0 cm depending on context)
If a number lacks a decimal point (e.g., 850000), trailing zeros may not be significant; adding a decimal point (850000.) or using scientific notation (8.5 × 10^5) clarifies sig figs
Example thinking about a measurement: a bolt between 1.7 and 1.8 cm with an instrument showing tenths and an estimated hundredths yields about 1.75 cm (three sig figs)
When the instrument’s precision differs (e.g., one ruler to 0.01 cm and another to 0.1 cm), the overall reported precision cannot exceed the least precise instrument

Align decimal places, then round to the least precise decimal place among the operands
Example: $6.2\quad(+) 9.203 \Rightarrow 15.4$ (tenths place)
For sums with different decimal places, round to the smallest decimal place present in any term

Use the e notation button (e or ee) or Times 10^x depending on your calculator
When combining, put coefficient and exponent in parentheses to avoid order-of-operations issues
Example workflow: compute $\frac{2.1 \times 10^{-3}}{100}$ by entering $2.1 \times 10^{-3}$ , then ÷ 100, with parentheses if needed

Zeros are not inherently significant; use the rules above
When converting to scientific notation, only include significant digits in the coefficient
The precision of results should reflect the least precise measurement in any calculation sequence