Geometric Bites

An improper fraction being turned into a mixed number. Fractions are used to represent parts of a whole, but once a fraction becomes larger than 1, there are two common ways to write it: as an improper fraction or as a mixed number . These two forms often represent exactly the same quantity; the difference is simply how the number is written. Improper fractions An improper fraction is a single fraction where the numerator is greater than or equal to the denominator. This means the fraction is at least 1 whole (and possibly more). Examples: 7/4 12/5 9/9 In 7/4, there are 7 parts, and each whole is made from 4 parts. Since 7 is larger than 4, the value is greater than 1. Mixed numbers A mixed number is written as a whole number followed by a proper fraction. The proper fraction shows the leftover part after counting whole units. Examples: 1 3/4 2 2/5 3 1/6 In 1 3/4, the “1” shows one whole, and “3/4” shows three extra quarters. Same value, differe...

Structure: The Great Pyramid of Giza, Egypt. Source:  https://commons.wikimedia.org/wiki/File:Kheops-Pyramid.jpg When physicists and mathematicians talk about the “architecture” of the universe, it can sound like they mean the universe was built like a house. Most of the time, they do not mean that. In this context, “architecture” is a practical word for something simpler and more specific: The universe behaves as if it has deep, consistent structure — stable rules and constraints that keep working wherever we look. This post explains why many scientists think that is a reasonable conclusion, without assuming any advanced mathematics. 1) The universe repeats itself: patterns that do not go away In everyday life, you already rely on the universe being stable: If you drop a mug, it falls. If you heat water, it boils at roughly the same temperature (given the same pressure). If you shine light through glass, it bends in predictable ways. Science extends this ...

A quadratic function and its turning point. Link to graph:  https://www.desmos.com/calculator/fktyfs12st A quadratic function is any function of the form f(x) = ax² + bx + c with a ≠ 0 . Its graph is a parabola, and every parabola has exactly one turning point (also called the vertex ). Completing the square is fundamental because it rewrites the quadratic as a shifted square , which makes the turning point immediately visible. Vertex form: the turning point is built in The vertex form of a quadratic is: f(x) = a(x − h)² + k This form is powerful because it exposes two facts at once: (x − h)² ≥ 0 for all real x (a square is never negative). (x − h)² = 0 happens exactly when x = h . So: If a > 0 , then a(x − h)² ≥ 0 , so the smallest possible value of f(x) is k , achieved at x = h (a minimum). If a < 0 , then a(x − h)² ≤ 0 , so the largest possible value of f(x) is k , achieved at x = h (a maximum). Therefore, in vertex form, the turning ...

Parabolas in sight: The Clifton Suspension Bridge, Bristol, United Kingdom. A quadratic function is a function whose graph is a parabola (a U-shaped curve). One of the most useful ways to write a quadratic is in vertex form , because it shows the parabola’s turning point immediately. 1) The vertex form A quadratic function in vertex form is written as: f(x) = a(x - h) 2 + k This form is especially helpful because the values h and k tell you the vertex directly. 2) The vertex (turning point) The vertex is the point where the parabola changes direction. In vertex form: Vertex = (h, k) If the parabola opens up , the vertex is the lowest point (a minimum). If the parabola opens down , the vertex is the highest point (a maximum). 3) What the number a does The number a controls two key things: the direction the parabola opens, and how wide or narrow it is. a > 0 means the parabola opens up (U-shape). a < 0 means the parabola opens dow...

1) n! (factorial) Meaning: tells you how many permutations (complete orderings) you can generate with n objects at your disposal. Definition: n! = n × (n−1) × (n−2) × … × 2 × 1, with 0! = 1. Example: 5! = 5×4×3×2×1 = 120. 2) n!/(n−r)! (permutations of r choices from n; order matters) Meaning: tells you how many ordered outcomes you can generate when you make r choices out of a collection of n objects, without reuse. How to see it: 1st choice: n options 2nd choice: (n−1) options 3rd choice: (n−2) options … rth choice: (n−r+1) options Multiply: n × (n−1) × … × (n−r+1) = n!/(n−r)!. Example (n=5, r=2): 5P2 = 5!/(5−2)! = 5!/3! = (5×4×3×2×1)/(3×2×1) = 5×4 = 20. 3) (n!/(n−r)!)/r! = n!/((n−r)! r!) (combinations; order neglected) Meaning: tells you how many selections you can make when choosing r objects from n, where order does not matter. ...

Geometric Bites has always been about clear diagrams and full derivations that help you see the structure of mathematics. Over time it has grown into a rich, freely accessible library of explanations, proofs and visual ideas. Geometry Insights is the premium companion to that work: a dedicated article site focused on GCSE, A-Level and Further Pure mathematics, written from first principles with carefully engineered diagrams and a long-term, structured archive in mind. Visit Geometry Insights: https://geometryinsights.wordpress.com What Makes Geometry Insights Different? Where Geometric Bites offers free posts and full derivations, Geometry Insights is a curated, subscription-only library. Each article is built to answer a deeper question: not just “how do I use this formula?” but “why does this formula exist at all?” Premium-only articles that go in depth on GCSE, A-Level and Further Pure topics. First-principles derivations that start from definitions and basic f...

The angle–difference identities are: sin(α − β) = sinα cosβ − cosα sinβ cos(α − β) = cosα cosβ + sinα sinβ They can be seen geometrically using a rectangle OZYX with a few right-angled triangles inside it. All side lengths can be written in terms of sinα, cosα, sinβ and cosβ. First, draw a right triangle OPQ with hypotenuse OQ = 1 and angle β at O. By definition: OP = cosβ (horizontal side), QP = sinβ (vertical side). Next, use OP as the hypotenuse of another right triangle OPZ. The right angle is at Z, and the angle at P is α. The hypotenuse is OP = cosβ, so: OZ = sinα cosβ, ZP = cosα cosβ. In a similar way, use QP as the hypotenuse of a right triangle PQY. The right angle is at Y, and the angle at Q is α. The hypotenuse is QP = sinβ, so: QY = cosα sinβ, PY = sinα sinβ. Drop a vertical line from Y to the base at Z, and a horizontal line from Y to the left side at X. This makes OZYX a rectangle with: base OZ, height ZY. On the...