Library

From Euclid's algorithm to integer solutions: fast GCD computation, Bézout's identity and theorem, Gauss's theorem and Euclid's lemma, and the complete solution of equations ax + by = c.

Prove step by step that the number e is not the quotient of any pair of integers, using its series expansion and Fourier's proof by contradiction.

Why light can knock electrons out of a metal — but only above a certain frequency. The experiment that established the idea of the photon and won Einstein his Nobel Prize.

Why some nuclei transform spontaneously by emitting radiation, how to write these decays, and the decay law that governs half-life and dating.

What if the security of a secret key no longer rested on hard computations, but on the laws of quantum physics? The BB84 protocol and eavesdropper detection.

Predicting how traits are passed on: dominant and recessive alleles, genotype and phenotype, Punnett squares and the 3:1 then 9:3:3:1 ratios, and the cases — codominance, sex linkage, polygeny — that go beyond Mendel's laws.

Why must good encryption stay public? Kerckhoffs's founding rule, the trap of security through obscurity, and how to design and evaluate a truly secure system.

Learn to speed up recursive algorithms that keep recalculating the same results, by memoizing what has already been computed. Tree and table diagrams illustrate the difference.

The experiment that gave birth to quantum physics in 1900: black-body radiation, the "ultraviolet catastrophe" of classical physics, and Planck's stroke of genius when he postulated that energy is exchanged in grains. With spectral curves and the laws of Wien and Planck.

How a gene becomes a working molecule: transcription into messenger RNA, the 64-codon genetic code, translation by ribosomes, and what a single changed base can do.

Explore the three main families of organic chemistry reactions — substitution, addition, elimination — through concrete examples involving alkenes and alcohols. Also learn to balance combustion equations for organic compounds, essential for understanding fuels.

The cell is the basic building block of every living being, from bacteria to humans. Discover its organization, its organelles, and what distinguishes an animal cell from a plant cell.

DNA is the molecule that carries, in each of your cells, all the genetic information passed down by your parents. Discover its double helix structure, the concept of genes and chromosomes, and how this information shapes your hereditary traits.

The species that inhabit the Earth today are just a snapshot of a history billions of years old. Discover Darwin's theory, the evidence for evolution, and how natural selection transforms populations, generation after generation.

How plants turn light into living matter: the balance 6 CO₂ + 6 H₂O → glucose + 6 O₂, the role of chloroplasts, the limiting factors, and why the oxygen we breathe comes from water.

How a cell reproduces: the cell cycle and DNA replication, mitosis that makes clones, meiosis that halves the chromosome number, and the shuffling that makes two children never quite alike.

How the body defends itself: barriers and innate immunity, the B and T lymphocytes of adaptive immunity, immune memory and vaccination, herd immunity, and the failures — allergies, autoimmunity, immunodeficiencies.

How information travels through the body: the anatomy of the neuron and the myelin sheath, the action potential and the all-or-none law, chemical transmission at the synapse, and from the reflex arc to brain plasticity.

How life organises into networks: trophic levels and the role of decomposers, the 10 % rule that limits food chains, the carbon and nitrogen cycles, and the trophic cascades that reshape whole landscapes.

How the body holds its constants steady: the hormonal message and its target cells, negative feedback, blood glucose regulation by insulin and glucagon, and what happens when the loop breaks — diabetes.

The strange heart of quantum physics: light sometimes behaves like a wave, sometimes like a stream of particles (photons), and matter itself has a wave nature. From the photoelectric effect to electron interference, with many diagrams.

Why atoms emit only certain very specific colors: quantized energy levels, the Bohr model, transitions and photons. Line spectra as the fingerprints of atoms, with level diagrams and calculations on the hydrogen atom.

One of the most profound ideas in physics: it is impossible to know both, with perfect precision, the position and the velocity of a particle. Not for lack of instruments, but by nature. Wave origin, the formulation ΔxΔp ≥ ħ/2, and consequences.

The concepts behind the modern quantum revolution: superposition of states, measurement that "makes a choice", Schrödinger's cat, the entanglement of two particles, and the qubit — the building block of the quantum computer. With diagrams and concrete examples.