Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
1,200 fundamentals
“The insulin that keeps millions alive is now brewed by bacteria we reprogrammed.”
PCR amplifies a DNA segment into many copies; gel electrophoresis separates DNA fragments by size using an electric field.
“The insulin that keeps millions alive is now brewed by bacteria we reprogrammed.”
A restriction enzyme cuts DNA at specific sites; DNA ligase joins DNA fragments together — the scissors and the glue.
“The insulin that keeps millions alive is now brewed by bacteria we reprogrammed.”
A vector (like a plasmid) carries the gene of interest; the host cell (like a bacterium) receives it and expresses the product.
“The insulin that keeps millions alive is now brewed by bacteria we reprogrammed.”
They recognise specific palindromic sequences and cut only at those exact sites.
“Only about 10% of energy passes up each step of a food chain — which is why lions are rare and grass is everywhere.”
Energy moves one-way through an ecosystem — from the sun to producers to consumers — and is lost as heat at each step.
Memory trick: nutrients cycle, but energy flows one way and dissipates as heat.
“Only about 10% of energy passes up each step of a food chain — which is why lions are rare and grass is everywhere.”
A frequent error is thinking energy is recycled like nutrients. In reality, energy moves one-way through an ecosystem — from the sun to producers to consumers — and is lost as heat at each step.
Memory trick: nutrients cycle, but energy flows one way and dissipates as heat.
“Only about 10% of energy passes up each step of a food chain — which is why lions are rare and grass is everywhere.”
Only about 10% of the energy at one trophic level passes to the next.
Memory trick: because only ~10% moves up, food chains are short and energy pyramids are upright.
“Only about 10% of energy passes up each step of a food chain — which is why lions are rare and grass is everywhere.”
A frequent error is thinking most energy is transferred up the food chain. In reality, only about 10% of the energy at one trophic level passes to the next.
Memory trick: because only ~10% moves up, food chains are short and energy pyramids are upright.
“Only about 10% of energy passes up each step of a food chain — which is why lions are rare and grass is everywhere.”
The feeding positions in a food chain: producers, primary consumers, secondary consumers, and decomposers.
Memory trick: decomposers close the loop by returning nutrients to the soil.
“Only about 10% of energy passes up each step of a food chain — which is why lions are rare and grass is everywhere.”
A frequent error is forgetting decomposers, which recycle nutrients back to producers. In reality, the feeding positions in a food chain: producers, primary consumers, secondary consumers, and decomposers.
Memory trick: decomposers close the loop by returning nutrients to the soil.
“Only about 10% of energy passes up each step of a food chain — which is why lions are rare and grass is everywhere.”
Real ecosystems are interlinked food webs rather than single isolated food chains.
Memory trick: a food web is many interconnected food chains — more realistic and stable.
“Only about 10% of energy passes up each step of a food chain — which is why lions are rare and grass is everywhere.”
A frequent error is assuming each organism eats only one type of food. In reality, real ecosystems are interlinked food webs rather than single isolated food chains.
Memory trick: a food web is many interconnected food chains — more realistic and stable.
“A vending machine is a function: one button, one snack, every time.”
A rule assigning each input exactly one output.
Memory trick: One input → exactly one output.
“You can't feed a negative into a square root over the reals — that limits the domain.”
Domain is the set of allowed inputs; range is the set of possible outputs.
Memory trick: Domain in, range out.
“A bijection (both) is a perfect pairing with no leftovers.”
A one-one (injective) function never repeats outputs; an onto (surjective) function hits every output.
Memory trick: Injective: no repeats; surjective: no gaps.
“Functions can be chained like an assembly line, each stage feeding the next.”
Applying one function to the result of another, (f∘g)(x) = f(g(x)).
(f∘g)(x)=f(g(x))
Memory trick: Work inside-out.
“Check the discriminant and you know the roots' nature before solving.”
For ax²+bx+c=0, D = b²−4ac decides the roots: D>0 two real, D=0 equal, D<0 complex.
D = b²−4ac
Memory trick: D<0 → complex roots (not 'none').
“One formula unlocks every quadratic ever written.”
The roots of ax²+bx+c=0 are x = [−b ± √(b²−4ac)]/2a.
x=(−b±√D)/2a
Memory trick: Memorise it cold.
“You can build a quadratic straight from its roots — no expansion needed.”
For ax²+bx+c=0, roots sum to −b/a and multiply to c/a.
Memory trick: Sum −b/a, product c/a.
“'Imaginary' numbers run the very real electronics in your phone.”
i is defined by i² = −1, extending numbers beyond the real line.
i² = −1
Memory trick: i² = −1.
“A complex number is just a point on a plane; its modulus is how far out it sits.”
|z| = √(a²+b²) is z's distance from the origin.
|z|=√(a²+b²)
Memory trick: Distance from origin.
“Multiplying by the conjugate is the trick to clear i from a denominator.”
The conjugate of a+bi is a−bi; z times its conjugate is |z|² (real).
z·z̄=|z|²
Memory trick: Flip the sign of i.
“Stacking equal steps — like saving a fixed amount every month.”
A sequence with a constant common difference; nth term a+(n−1)d.
aₙ=a+(n−1)d
Memory trick: Add a fixed step each time.
“Compound interest and viral sharing both grow geometrically.”
A sequence with a constant ratio; nth term a·r^(n−1).
aₙ=a·r^(n−1)
Memory trick: Multiply by a fixed ratio.