§1Rearranging formulas
Transposition is undoing: whatever was last done to the wanted symbol is removed first, applying the same operation to both sides until the symbol stands alone.
| To remove… | Apply to both sides… |
|---|---|
| + a / − a | subtract a / add a |
| × a / ÷ a | divide by a / multiply by a |
| square / square root | take the square root / square |
| sin, cos, tan | arcsin, arccos, arctan |
| eˣ / ln x | take ln / exponentiate |
The sine bar obeys H = L sin θ. To find θ from a measured stack: divide by L, then undo the sine: θ = arcsin(H/L). A 61.538 mm stack under a 250 mm bar gives θ = arcsin 0.246 15 = 14°15′ — the Solution of Triangles page run backwards.
First, do one operation per line — speed is where sign errors breed. Second, finish with a units check: if the rearranged formula’s units don’t reduce to the units of the answer, the algebra is wrong somewhere, guaranteed.
§2Essential identities and factoring
A handful of expansion patterns, read in both directions, do most of algebra’s heavy lifting: expand to simplify, factor to solve.
| Name | Identity |
|---|---|
| Square of a sum | (a ± b)² = a² ± 2ab + b² |
| Difference of squares | a² − b² = (a − b)(a + b) |
| Cube of a sum | (a ± b)³ = a³ ± 3a²b + 3ab² ± b³ |
| Sum / difference of cubes | a³ ± b³ = (a ± b)(a² ∓ ab + b²) |
The difference of squares earns its place in numerical work: computing c² − a² as (c − a)(c + a) preserves significant figures when a is close to c — the reason the right-triangle solution table on the Solution of Triangles page uses the factored form.
Contents§3Linear and simultaneous equations
One unknown, first power: isolate it. Two unknowns need two independent equations; eliminate one variable and the problem collapses to the first kind.
Stock A is 10 % tin, stock B is 20 % tin; 50 kg of 16 % alloy is wanted. With x kg of A and y of B: x + y = 50 and 0.10x + 0.20y = 8. Multiply the first by 0.10 and subtract: 0.10y = 3, so y = 30 kg of B, x = 20 kg of A. Check: 2 + 6 = 8 kg of tin — 16 % of 50.
Three or more unknowns solve the same way in principle, but the bookkeeping is better done with determinants and matrix inverses — the Matrices page carries this exact example style through Cramer’s rule.
Contents§4Quadratic equations
Any equation reducible to ax² + bx + c = 0 surrenders to one formula; the discriminant under the root tells the story before you finish.
| Sign of D | Roots |
|---|---|
| D > 0 | two distinct real roots |
| D = 0 | one repeated real root (the parabola grazes the axis) |
| D < 0 | a complex conjugate pair — no real crossing |
3x² − 5x − 7 = 0: D = 25 + 84 = 109, so x = (5 ± √109)/6 = 2.573 or −0.907. In a physical problem one root usually fails a reality test (a negative length, an impossible angle) — compute both, then discard with a stated reason, never silently.
§5Cubic and higher equations
Cubics have a closed formula, but almost nobody should use it. Know what the algebra guarantees, then reach for a numerical method.
Every cubic with real coefficients has at least one real root; substituting x = t − a/3 (for x³ + ax² + bx + c) removes the square term and leaves a depressed cubic t³ + pt + q = 0, which Cardano’s radicals can crack — at the price of complex intermediate arithmetic even when all three roots are real. For quartics the situation is worse, and beyond degree four no radical formula exists at all.
Sketch or tabulate the polynomial to bracket each real root with a sign change, then polish with Newton–Raphson (§6). Two iterations from a decent bracket beat the closed formula for speed and for arithmetic safety.
§6Newton–Raphson root finding
Slide down the tangent: from a guess xₙ, follow the tangent line of f to where it crosses zero, and call that the next guess. Convergence is ferociously fast near a simple root.
f(x) = x³ − 17, f′(x) = 3x². From x₀ = 2.5: x₁ = 2.5 − (−1.375)/18.75 = 2.573 333; x₂ = 2.571 283; x₃ = 2.571 282 — six decimals in three steps. (∛17 = 2.571 281 6.)
Gear work often needs φ from inv φ = tan φ − φ (Solution of Triangles, §10) — no algebraic inverse exists. For inv φ = 0.030 000, take f(φ) = tan φ − φ − 0.03, f′(φ) = tan²φ, start at 25°: the first step lands on 25.0065° and the second confirms it — φ = 25°00′ to the minute. Interpolation in a table gets close; Newton–Raphson gets exact.
Near a flat spot (f′ ≈ 0) the tangent shoots to the horizon, and with a poor start the iteration can cycle or diverge. Bracket the root first with a sign change; if an iterate leaves the bracket, bisect once and resume.
§7Arithmetic and geometric progressions
An arithmetic progression climbs by adding; a geometric one by multiplying. Machine tools are built on the second: speed steps feel even to the operator only when each is a constant ratio above the last.
Gauss’s schoolroom trick is the AP sum in action: 1 + 2 + ··· + 100 = 100 × 101/2 = 5050.
A gearbox must span 45 to 1400 rev/min in six geometric steps. The ratio is r = (1400/45)¹ᐟ⁵ = 1.989, giving 45, 89.5, 178, 354, 704, 1400 rev/min. Each step raises speed by the same percentage, so cutting-speed coverage is even across the whole diameter range — the reason speed plates on real machines read like this.
§8The binomial shortcut
For small x, powers linearise: the first two terms of the binomial series are usually all an estimate needs.
(1.002)¹⁰ ≈ 1 + 10 × 0.002 = 1.020; the exact value is 1.020 18. The neglected term ½(10)(9)(0.002)² = 0.000 18 predicts the shortfall exactly — the approximation comes with its own error bar.
The same move gives the workshop staples √(1 + x) ≈ 1 + x/2 and 1/(1 + x) ≈ 1 − x — good to better than 0.1 % whenever |x| < 0.05.
Contents§9Derivatives
The derivative is instantaneous rate of change — slope, velocity, sensitivity. Its killer application in design work: where a smooth quantity peaks, its derivative is zero.
| f(x) | f′(x) | f(x) | f′(x) |
|---|---|---|---|
| xⁿ | n xⁿ⁻¹ | eˣ | eˣ |
| sin x | cos x | ln x | 1/x |
| cos x | −sin x | u·v | u′v + uv′ |
| tan x | sec²x | u/v | (u′v − uv′)/v² |
| f(g(x)) | f′(g)·g′ (chain rule) | c·f | c·f′ |
| Angles in radians throughout — the trigonometric derivatives are only this clean in radian measure. | |||
Squares of side x are cut from the corners of a 400 × 250 mm sheet and the sides folded up: V = x(400 − 2x)(250 − 2x). Setting dV/dx = 12x² − 2600x + 100 000 = 0 gives x = 50 or 166.7 mm; only x = 50 is physical (166.7 exceeds half the width). x = 50 mm, V = 50 × 300 × 150 = 2.25 litres — and because dV/dx = 0 there, a millimetre of error in x costs almost nothing, which is itself useful to know.
§10Integrals
Integration is accumulation — area under a curve, work from a varying force, volume from stacked slices. It runs the derivative table in reverse.
| f(x) | ∫ f dx | f(x) | ∫ f dx |
|---|---|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) | eˣ | eˣ |
| 1/x | ln |x| | sin x | −cos x |
| cos x | sin x | sec²x | tan x |
A spring of rate k = 18 N/mm obeys F = kx. Work to compress it 40 mm is the accumulated force: W = ∫₀⁴⁰ kx dx = ½k x² = ½ × 18 × 1600 = 14 400 N·mm = 14.4 J. The ½ is the triangle under the F–x line — integration formalising what the area picture already says.
Definite integrals evaluate the antiderivative at the limits and subtract: the constant C cancels. Where no antiderivative exists in closed form — the ellipse perimeter on the Geometry page is the classic case — numerical integration steps in without complaint.
Contents§11Quick reference
The working core of the page on one card rack.
Quadratic
x = (−b ± √(b²−4ac))/2a
D = b² − 4ac decides the roots
Newton–Raphson
x₊ = x − f(x)/f′(x)
bracket first, then slide
Progressions
AP: Sₙ = n(a₁+aₙ)/2
GP: aₙ = a₁rⁿ⁻¹
Binomial
(1+x)ⁿ ≈ 1 + nx
√(1+x) ≈ 1 + x/2
Derivative
d(xⁿ)/dx = nxⁿ⁻¹
max/min where f′ = 0
Integral
∫xⁿdx = xⁿ⁺¹/(n+1)
work = ∫F dx