Basics of Math
The language engineers use to describe and predict the world. You need less of it than you think — just the ideas behind the symbols.
A ratio compares two amounts: 3 parts fiber to 1 part resin. A percentage is a ratio out of 100. 'Fiber volume fraction 60%' means 60 of every 100 units of volume is fiber.
When one thing grows in step with another. Double the pull, double the stretch — up to a point. The steepness k is how much y changes for each step of x.
A shorthand for very big or very small numbers. 10³ = 1,000; 10⁻⁶ = one millionth. A carbon fiber is about 7×10⁻⁶ m across — 7 micrometers, thinner than a hair.
Add the values, divide by how many there are. If you measure a part's thickness at five spots, the average smooths out the bumps into one representative number.
How fast something changes. On a stress–strain graph, a steep slope means a stiff material — it takes a lot of stress to stretch it a little. This slope is the modulus.
Reading between known points. If a material is rated at 20°C and 100°C, you can estimate its behavior at 60°C by assuming it changes smoothly in between.
A number means nothing without its unit — 5 could be metres, seconds or apples. Engineers keep the unit attached to every number and convert carefully: 1 metre is 100 centimetres, 1 GPa is 1000 MPa. Getting a unit conversion wrong has crashed spacecraft — literally.
Make something twice as wide and its area grows four times (2×2) and its volume eight times (2×2×2). This 'square–cube law' is why big animals have thick legs, and why scaling a part up is never as simple as it looks — the weight races ahead of the strength.
A scale where each step multiplies instead of adds — 1, 10, 100, 1000. It squeezes an enormous range onto one chart, so you can show a bacterium and a whale together. Engineers use it constantly: fatigue life, material selection charts and the periodic trends here all use log scales.