Numbers, Fractions, and Decimals

§1Signed numbers and order of operations

Negative numbers carry direction — a coordinate left of datum, a shortfall, a temperature drop. The sign rules and a fixed order of evaluation keep long expressions unambiguous.

Expressions evaluate in a fixed order — brackets first, then indices (powers and roots), then multiplication and division left to right, then addition and subtraction left to right. So 5 + 3 × 2² = 5 + 12 = 17, not 32; brackets override: (5 + 3) × 2² = 32. When transcribing a formula into a calculator or spreadsheet, brackets are cheaper than a scrapped part — use more than the minimum.

§2Ratio and proportion

A ratio compares two quantities of the same kind; a proportion states that two ratios are equal. Most gearing, scaling and mixing arithmetic is proportion in light disguise.

Quantities are in direct proportion when they rise together (cutting time and length of cut) and in inverse proportion when one rises as the other falls (gear teeth and speed, spindle speed and diameter at constant surface speed).

Compound proportion chains several factors: if a job’s time is proportional to length and inversely proportional to feed, doubling the length and increasing feed by half changes time by 2 ÷ 1.5 = 1.33 — a third longer, no fresh algebra required.

§3Percentage

A percentage is a fraction with denominator 100 — nothing more. Keep clear what the 100 % base is, and the rest is multiplication.

§4Common fractions and reciprocals

Workshop fractions survive because inch tooling still arrives in 64ths. The arithmetic never changes: common denominators to add, invert to divide.

Reciprocals turn division into multiplication — useful when one divisor recurs. Dividing repeatedly by 0.039 37 (the old inch-per-millimetre figure) is the same as multiplying by its reciprocal 25.400 1; the modern definition below makes the exact factor 25.4.

§5Decimals and the inch–millimetre bridge

Since 1959 the inch has been defined as exactly 25.4 mm, so every conversion below is exact arithmetic, not measurement.

A decimal fraction terminates only when its denominator contains no prime factors other than 2 and 5 — which is why every 64th of an inch has an exact decimal, and why thirds recur. The table gives all sixty-four steps with their exact millimetre equivalents.

§6Continued and conjugate fractions

When a machine offers only whole-number gear teeth, the problem becomes: find a fraction with small terms that approximates a given decimal as closely as possible. Continued fractions solve it optimally.

Any number unfolds into a staircase of whole parts and reciprocals. Cutting the staircase early yields the convergents — the best possible fractions for their size of denominator:

Conjugate fractions are pairs a/b and c/d with ad − bc = ±1; each is the closest fraction to the other for its denominator, and their mediant (a+c)/(b+d) lands between them. Successive convergents above are conjugate — the property that guarantees their optimality, and a quick test when hunting change-gear ratios by hand.

§7Powers, roots and powers of ten

Index laws compress big arithmetic; powers-of-ten notation keeps the decimal point honest when the numbers span galaxies or microns.

In powers-of-ten form every value is a number between 1 and 10 times a power of ten. Multiply the leading numbers, add the exponents; divide, subtract them: (6.4 × 10³)(2.5 × 10⁻⁵) = 16 × 10⁻² = 0.16, and (8.4 × 10⁵) ÷ (2.4 × 10⁻²) = 3.5 × 10⁷.

§8Logarithms

A logarithm answers “ten (or e, or two) to what power gives this number?” — turning multiplication into addition and powers into products, which is why they run growth, decay and decibel arithmetic.

Common logs (base 10) suit orders of magnitude: log 250 = 2.398 — a number of magnitude 10². Natural logs (base e) suit anything growing at a rate proportional to itself: ln 250 = 5.521. Base 2 counts doublings: log₂ 10 = 3.32, so a thousand ≈ ten doublings.

§9Imaginary and complex numbers

Define i so that i² = −1 and every polynomial gains a full set of roots — and, more usefully for engineers, every rotating or oscillating quantity gains a one-line algebra.

A complex number z = a + bi plots as the point (a, b). Its polar form carries the same information as a length and a direction:

Adding complex numbers adds their components, like vectors. Multiplying them multiplies the lengths and adds the angles — multiplication is rotation-and-scale, the fact that makes phasor analysis of vibration and AC circuits work.

§10Factorials, permutations and combinations

Counting arrangements is the doorway to probability and to the statistics of §Manufacturing Data Analysis. Three formulas cover it.

Permutations count ordered selections; combinations count selections where order is irrelevant. 10! = 3 628 800 — factorials outgrow everything, which is why counting problems turn astronomical so quickly.

§11Prime numbers and factors

Primes are the atoms of the whole numbers. Factorising exposes what a number can be evenly divided into — the question behind gear trains, indexing plates and batch splitting.

To factorise, divide out the smallest primes in turn: 3960 = 2 × 1980 = 2² × 990 = 2³ × 495 = 2³ × 3 × 165 = 2³ × 3² × 55 = 2³ × 3² × 5 × 11. Any usable gear pair, index step or even subdivision of 3960 must assemble from those atoms.

§12Constants worth memorising

A dozen numbers appear so often that fetching them from memory is faster than any lookup. Computed here to six figures.

§13Quick reference

The working core of the page on one card rack.