In short

(a - b)^2 is a picture, not a formula. See a big a \times a square. Peel off two a \times b strips (one from the right, one from the bottom) — that is the -2ab. The little b \times b corner where the strips overlapped got peeled twice, so paste it back — that is the +b^2. The formula a^2 - 2ab + b^2 is what your eyes read off the picture. You do not memorise it; you see it.

You are sitting in an exam at 10:47 am. Question 4(b) wants (2x - 3)^2 expanded. Or it wants 98^2 in your head with no calculator. You have maybe five seconds before you commit to an answer. Five seconds is not enough to derive the identity from first principles, and it is just enough time for your hand to write a^2 - 2ab - b^2 — wrong sign on the last term — if you are working from rote memory under stress.

What you want instead is a thumbnail — a picture so small and clean that it lives in the corner of your mind and pops up the instant you see (\text{something} - \text{something})^2. The full geometric proof lives in the sibling article. This article is just the thumbnail.

The thumbnail

Mental thumbnail for (a minus b) squared A small square divided into four regions: a large top-left region labelled (a-b)squared, two thin shaded strips on the right and bottom each labelled minus ab, and a tiny darker square in the bottom-right corner labelled plus b squared. (a−b)² −ab −ab +b² a² −2ab +b²
The thumbnail: $a^2$ minus two $ab$ strips, plus the over-peeled $b^2$ corner. That is the entire identity, in one glance.

Three regions to remember:

  1. The whole square is a^2. This is your starting canvas.
  2. The two shaded strips are each ab, and they are gone. Subtract 2ab. Why two? You needed to peel from the right and the bottom to shrink the square in both directions. One strip would only shrink one side.
  3. The dark little corner is b^2, and it gets added back. Why added, not subtracted? Because both strips passed through that same little corner, so when you subtracted the two strips you removed the corner twice. Once is correct; the second removal must be undone, so you add b^2 back.

That is it. Three numbers — a^2, -2ab, +b^2 — and they fall out of the picture in the order your eye scans it. You do not check a formula sheet; you check your thumbnail.

How to use the thumbnail in 5 seconds

When you see (\text{thing}_1 - \text{thing}_2)^2, do not reach for the formula. Do this instead:

  1. Call the first thing a, the second thing b.
  2. Picture the big square. Write down a^2.
  3. Picture the two strips. Write down -2ab.
  4. Picture the corner. Write down +b^2.

Three glances. Three terms. The hardest part is not getting confused about whether the corner is +b^2 or -b^2 — and the picture solves that for free, because in your thumbnail the dark corner is always the one you paste back.

Example 1 — $99^2$ in your head

You see 99^2. Rewrite as (100 - 1)^2, so a = 100 and b = 1. Now look at your thumbnail.

  • Big square: 100 \times 100 = 10000.
  • Two strips: each 100 \times 1 = 100. Together 200. Subtract.
  • Corner: 1 \times 1 = 1. Add back.
99^2 = 10000 - 200 + 1 = 9801

Why this is automatic: the picture told you what to compute, in what order, and what sign to use. There was nothing to remember.

This is exactly the ekādhikena pūrveṇa shortcut from Vedic mathematics, dressed in geometric clothing. Bhāratī Kṛṣṇa Tīrtha's sūtras describe squaring numbers near a base (like 100) using complements, and the strips-and-corner picture is the reason the trick works.

Example 2 — $98^2$ in your head

98 = 100 - 2, so a = 100 and b = 2. Glance at the thumbnail.

  • Big square: 10000.
  • Two strips: each 100 \times 2 = 200. Together 400. Subtract.
  • Corner: 2 \times 2 = 4. Add back.
98^2 = 10000 - 400 + 4 = 9604

You just squared a two-digit number while your friend was still hunting for their calculator. Why this beats long multiplication: long multiplication needs you to handle carries across four digit-pair products. The thumbnail needs three additions where two of them are essentially "shift a zero". Far less working memory.

Cricket twist: the run rate after 98 overs in a hypothetical innings, squared for some imagined statistic, is 9604. You did not write a single intermediate step.

Example 3 — $(2x - 3)^2$, the algebra version

This is where students slip and write something like 4x^2 - 6x + 9 or 4x^2 - 12x - 9. The thumbnail does not slip. Set a = 2x and b = 3.

  • Big square: a^2 = (2x)^2 = 4x^2.
  • Two strips: each a \cdot b = (2x)(3) = 6x. Together 12x. Subtract.
  • Corner: b^2 = 3^2 = 9. Add back.
(2x - 3)^2 = 4x^2 - 12x + 9

Why the strips are 12x and not 6x: the picture has two strips, not one. The "2" in 2ab is not optional decoration — it is literally the count of strips you peeled off. If you only see one strip in your mind, you will drop a factor of two.

Why the picture survives where the formula breaks

A formula is a string of symbols. Under exam stress, strings of symbols mutate: signs flip, factors of two vanish, the last term becomes negative. A picture mutates much less, because each region of the picture has a physical reason to be there — a strip you peeled, a corner you double-counted. Reasons are stickier than symbols.

This is also why the picture transfers. The same "subtract too much, then add the overlap back" logic shows up in the inclusion–exclusion principle for counting, in signed-area arguments for polygons, and even in the way physicists handle overlapping electric fields. You have not just learned an algebra identity; you have learned a habit of thought.

The thumbnail in the wild

Numbers near a round base — 19, 48, 97, 199, 498, 999 — all yield to the thumbnail in seconds:

Number Rewrite Big − strips + corner Answer
19^2 (20 - 1)^2 400 - 40 + 1 361
48^2 (50 - 2)^2 2500 - 200 + 4 2304
97^2 (100 - 3)^2 10000 - 600 + 9 9409
199^2 (200 - 1)^2 40000 - 400 + 1 39601
499^2 (500 - 1)^2 250000 - 1000 + 1 249001

Each row is a separate thumbnail-glance. None required pen-and-paper multiplication. Why this works for every near-base number: the identity is true for all a and b, and the thumbnail is just the identity made visible. Rounding to a nearby base is a choice of a — the picture does not care which base you picked.

That is the whole point of an identity-as-picture: you carry one thumbnail in your head, and it pays rent every time you see a square that is "close to a round number" or an algebraic expression of the form (\text{something} - \text{something})^2.

References

  1. Algebraic identities — the parent article with all eight standard identities.
  2. Geometric proof of (a − b)² — the full sibling proof, with the over-counting argument spelled out.
  3. Wikipedia: Vedic Mathematics — the ekādhikena pūrveṇa sūtra and other base-complement squaring shortcuts.
  4. NCERT Class 8 Mathematics, Chapter 9: Algebraic Expressions and Identities — the official syllabus treatment of the square identities.
  5. Cut the Knot: Proofs without words — a library of identity-as-picture proofs in the same spirit.