- 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 uncertainty is half of the smallest scale. (eg if a ruler can measure to $0.1cm$, the uncertainty would $0.005 cm$)
- 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.
Vectors are drawn as arrows whose length represents their scale/magnitude and their orientation refer to their direction. A variable representing a vector is written with a right-pointing arrow above it.
Notice that when we are subtracting vectors ($\vec{a} - \vec{b}$), we have **tail** to **tail** and the difference vector has a direction from ($\vec{b} to \vec{a}$)
Vectors can be broken up into two vectors (**"components"**) laying on the x- and y-axes via trigonometry such that the resultant of the two components is the original vector. This is especially helpful when adding larger (3+) numbers of vectors.
The product of a vector multiplied by a scalar is a vector with a magnitude of the vector multiplied by the scalar with the same direction as the original vector.
A **scientific model** is a simplification of a system based on assumptions used to explain or make predictions for that system.
!!! note "Definition"
- **System**: An object or a connected group of objects.
- **Point particle assumption**: An assumption that models a system as a blob of matter. It is more reliable if the size and shape of the object(s) do not matter much.
- **Uniform motion**: The type of motion in which the speed of an object is constant.
### Displaying motion
Motion can be expressed visually using a **motion diagram** or a **position-time graph**.
// TODO: insert motion diagram here because kognity bad
A **position-time graph** expands on the motion diagram by specifying a precise **position** value on the vertical axis in addition to time on the horizontal axis. The line of best fit indicates the object's speed, as well as if it is accelerating or decelerating.
