Drawing ray diagrams tells us the nature of an image, but to find exact distances we need numbers and a consistent sign system. The New Cartesian Sign Convention fixes the rules. The pole of the mirror is taken as the origin and the principal axis as the x-axis. All distances are measured from the pole. Distances measured against the direction of incident light (i.e. to the left, behind the mirror in the usual diagram) are negative, while distances measured along the incident light are positive. Heights measured upwards (above the axis) are positive and those measured downwards are negative.
Because the object is always placed in front of a mirror, the object distance $u$ is always negative. A concave mirror has a negative focal length; a convex mirror has a positive focal length. These signs are not optional — putting in the wrong sign is the single most common mistake in numericals.
The mirror formula links the object distance, image distance and focal length:
- $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$, where $v$ is the image distance, $u$ is the object distance and $f$ is the focal length.
The magnification (m) tells us how large the image is compared with the object, and whether it is erect or inverted. It is defined as the ratio of image height $h'$ to object height $h$, and is also related to the distances:
- $m = \frac{h'}{h} = -\frac{v}{u}$.
The sign of $m$ carries meaning. A negative magnification means the image is real and inverted; a positive magnification means the image is virtual and erect. If $|m| > 1$ the image is enlarged, if $|m| < 1$ it is diminished, and if $|m| = 1$ it is the same size as the object. To solve a numerical, first write down every known quantity with its correct sign, substitute into the mirror formula to find the unknown distance, and then use the magnification relation. Always interpret the final signs to describe the image fully — never report a bare number without saying whether the image is real or virtual, erect or inverted.