- The leftmost non-zero digit is the **most significant digit**.
- If there is no decimal point, the rightmost non-zero digit is the **least significant digit**.
- Otherwise, the right-most digit (including zeroes) is the least significant digit.
- All digits between the most and least significant digits are significant.
- Pure (discrete) numbers are unitless and have infinite significant figures.
!!! example
In $123000$, there are 3 significant digits.<br>
In $0.1230$, there are 4 significant digits.
- When adding or subtracting significant figures, the answer has the **same number of decimals** as the number with the lowest number of decimal points.
- When multiplying or dividing significant figures, the answer has the **same number of significant figures** as the number with the lowest number of significant figures.
- Values of a calculated result can be **no more precise** than the least precise value used.
When rounding an answer with significant figures, if the **least significant figure** is $5$, round up only if the **second-least** significant figure is **odd**.
Scientific notation is written in the form of $m×10^{n}$, where $1 \leq m <10,n \in \mathbb{Z}$.Alldigitsbeforethemultiplicationsigninscientificnotationaresignificant.<br>
!!! example
The speed of light is 300 000 000 ms<sup>-1</sup>, or $3×10^8$ ms<sup>-1</sup>.
### Orders of magnitude
The order of magnitude of a number can be found by converting it to scientific notation and taking its power of 10.
| Caused by imperfect measurements and is present in every measurement. | Caused by a flaw in experiment design or in the procedure. |
| Can be reduced (but not avoided) by repeated trials or measurements. | Cannot be reduced by repeated measurements, but can be avoided completely. |
| Error in precision. | Error in accuracy. |
!!! example
- The failure to account for fluid evaporating at high temperatures is a systematic error, as it cannot be minimised by repeated measurements.
- The addition of slightly more solute due to uncertainty in instrument data is a random error, as it can be reduced by averaging the result of multiple trials.
Uncertainties are stated in the form of [value] ± [uncertainty]. A value is only as precise as its absolute uncertainty. Absolute uncertainty of **measurement** is usually represented to only 1 significant digit.
!!! note
Variables with uncertainty use an uppercase delta for their uncertainty value: $a ± \Delta a$
- The absolute uncertainty of a number is written in the same unit as the value.
- The percentage uncertainty of a number is the written as a percentage of the value.
!!! example
- Absolute uncertainty: 1.0 g ± 0.1 g
- Percentage uncertainty: 1.0 g ± 10%
To determine a measurement's absolute uncertainty, if:
- the instrument states its uncertainty, use that.
- an analog instrument is used, the last digit is estimated and appended to the end of the reported value. The estimated digit is uncertain by 5 at its order of magnitude.
- a digital instrument is used, the last reported digit is uncertain by 1 at its order of magnitude.
!!! example
- A ruler has millimetre markings. A pencil placed alongside the ruler has its tip just past 14 mm but before 15 mm. The pencil is 14.5 mm ± 0.5 mm long.
- A digital scale reads 0.66 kg for the mass of a human body. The human body has a mass of 0.66 kg ± 0.01 kg.
!!! info
See [Dealing with Uncertainties](/resources/g11/physics-uncertainties.pdf) for how to perform operations with uncertainties.
### Error bars
Error bars represent the uncertainty of the data, typically representing that data point's standard deviation, and can be both horizontal or vertical.
