Sign of the Times
The lens formula with its signs — where the numbers come from
What this lesson covers
Why it matters
Half the marks lost on lens numericals are ONE missing minus sign. The convention isn't decoration — it's what tells you real from virtual before you draw anything.
The idea in plain words
The chapter's master equation. Tap each piece.
1/v − 1/u = 1/f · m = v/u
Object 15 cm from a convex lens of f = 10 cm. Find v and m.
Real, inverted image at 30 cm, twice the size (m = −2)
- cm, with sign: u — object distance — NEGATIVE for a real object (it sits against the light direction)
- cm, with sign: v — image distance — positive = real (far side), negative = virtual (same side)
- cm, with sign: f — focal length — positive for convex, negative for concave
- ratio: m — magnification v/u — negative m means inverted image
- 1/v − 1/(−15) = 1/10
- 1/v = 1/10 − 1/15 = 1/30 → v = +30 cm (real)
- m = v/u = 30/(−15) = −2
Predict first
Using the convention (distances from the lens, direction of light = positive), a VIRTUAL image has v…
Virtual images sit on the object's side, against the light direction → v < 0. The sign IS the physics.
- negative — it forms on the same side as the object — correct
- positive — all images are positive
- zero
What you do
Drive the formula: park the image exactly at 2F, then push the object inside F and watch v flip negative.
Check yourself
For ANY lens with a real object, u is taken as…
The object sits on the incoming-light side, opposite to the positive direction — u < 0 regardless of the lens.
A calculation gives m = −3. The image is…
The sign carries orientation: negative m = inverted (hence real, for a single lens); |m| = 3 = size ratio.
For a concave lens, f is negative because…
Signs encode geometry: the diverging lens's focus sits against the light direction → negative.
- negative — correct
- positive
- negative only for concave lenses
- real, inverted, three times larger — correct
- virtual and diminished
- impossible — m can't be negative
- its focus is virtual, on the incoming side — correct
- it makes images smaller
- convention is arbitrary