Units and notation: why there are so many

A unit is an agreement

How long a metre is was not set by nature but agreed by people. Different eras and regions made different agreements, and those survive side by side today. Conversion exists not because the arithmetic is hard but because people working from different agreements read the same document. Even the international metric system has been redefined several times since it was first set, and today its values are pinned to physical constants such as the speed of light. The agreement has become more precise, but it is still an agreement.

Where conversions go wrong

Errors usually come from assumptions rather than formulas. Units sharing a name while differing in value, mixing weight with volume, and treating a scale whose zero is not absolute as a ratio are the common cases. A calculator will not catch these mistakes for you. It processes the numbers you give it exactly, but it never asks which region's gallon you meant.

• Same name, different value: cups, gallons and ounces differ by region
• Mixing weight and volume: a cup of flour weighs differently by how it is filled
• Scales without a true zero: Celsius and Fahrenheit cannot be doubled
• Prefix magnitudes: data sizes mix 1000 and 1024

There are several ways to write a number

The same value can be written in decimal, binary or hexadecimal. None is correct in itself; each suits a different purpose. Colours are written in hexadecimal, permissions in octal and flags in binary because those notations fit those jobs best. This is where notation differs from unit conversion: the value itself never changes. Switching bases is less like converting and more like changing clothes.

Letters and colours are numbers too

Inside a computer, characters and colours are stored as numbers. The agreement about which number means which character is a character encoding, and the agreement about which number means which colour is a colour space. Broken text and colours that look wrong almost always come from two sides using different agreements. That is why this collection treats in-computer agreements such as number bases, character encodings, colour notation and data sizes with the same eye as physical units like length, weight and temperature.

A habit of checking

When a conversion looks wrong, start with the order of magnitude. Being off by around a thousand suggests a prefix, by around double suggests a definition such as radius versus diameter, and by a fixed addition suggests a zero point. Knowing a rough value in advance tells you immediately when a result is wrong. For one more check, convert the result back and see whether you arrive at the original value.

How to read this collection

If you are new, start with unit systems and temperature to learn the traps in physical units, then read number bases, and move on to character encoding, colour notation and data sizes. Those last three are far easier once you know number bases, especially hexadecimal and powers of two.

• Unit systems: metric and imperial, same names with different values
• Temperature: converting scales with different zero points
• Number bases: why binary and hexadecimal are used
• Encoding, colour and data size: computer agreements built on bases

Reference values worth memorising

You do not need every conversion factor, but a handful lets you judge at once whether a result is right. The values below are either exact by definition or widely used references.

• 1 inch = 2.54 cm, 1 foot = 30.48 cm, 1 mile ≈ 1.609 km
• 1 pound ≈ 0.4536 kg, 1 ounce (weight) ≈ 28.35 g
• 0 °C = 32 °F, 100 °C = 212 °F, 0 K = −273.15 °C
• 1 byte = 8 bits, 1 KiB = 1024 bytes, 1 GiB ≈ 1.074 GB

What to check when using a tool

A converter removes arithmetic slips, but it does not choose your assumptions. Before typing, decide which region's definition the original value follows, whether it is weight or volume, and whether it is a value or a difference. Keep only as many decimal places as you need, and do not pretend to more precision than the source had. Writing 'about 5 miles' as 8.04672 km invents accuracy that was never there.