Geometric Bites

This article presents the rules of logarithms using complete, line-by-line derivations. Every identity is built directly from its exponential origin, without shortcuts, matching the structure of formal handwritten algebra. 1. Definition We begin with fundamental exponent facts: a⁰ = 1 ⇒ logₐ(1) = 0 a¹ = a ⇒ logₐ(a) = 1 Say: aᵐ = p Then, by definition: logₐ(p) = m Raise both sides of aᵐ = p to the power 1/m (with m ≠ 0 ): p^(1/m) = a Therefore: logₚ(a) = 1/m Since m = logₐ(p) , we obtain: logₐ(p) = 1 / logₚ(a) 2. Product Rule — Full Derivation Say: aᵐ = p and aⁿ = q Multiply: aᵐ · aⁿ = p · q Using index addition: a^(m+n) = p · q Taking logarithms: logₐ(p · q) = m + n Substitute: logₐ(p · q) = logₐ(p) + logₐ(q) 3. Quotient Rule — Full Derivation Say: aᵐ = p and aⁿ = q Divide: aᵐ / aⁿ = p / q Index subtraction gives: a^(m−n) = p / q Taking logarithms: logₐ(p / q) = m − n So: log...

Arithmetic and Geometric Sequences Arithmetic and geometric sequences are two fundamental types of numerical progressions. They describe how quantities grow or shrink by addition or by multiplication, and they form the foundation for topics such as series, summation formulas, and exponential growth. 1. Arithmetic Sequence An arithmetic sequence is a list of numbers in which each term differs from the previous one by a fixed amount called the common difference d . a, a + d, a + 2d, a + 3d, … , a + (n − 1)d a – first term d – common difference The n th term, denoted T n , is given by: T n = a + (n − 1)d Each new term is obtained by adding d to the previous term. The difference between consecutive terms remains constant: T k+1 − T k = d Example: If a = 4 and d = 3, the sequence is 4, 7, 10, 13, 16, … The 20th term is T 20 = 4 + (20 − 1)×3 = 61. 2. Geometric Sequence A geometric sequence is a list of numbers where each term is fo...

What Is an Isomorphism? In mathematics, an isomorphism is a function that shows two mathematical objects have the same structure. Although the objects may look different, an isomorphism demonstrates that they behave in exactly the same way with respect to the operations that define them. If such a map exists, the objects are called isomorphic . An isomorphism tells us that two systems are essentially the same, differing only by a relabelling of their elements. The Basic Idea An isomorphism is a function: f : A → B that must be: Injective — different elements of A map to different elements of B. Surjective — every element of B comes from some element of A. Together these mean f is bijective , and so it has an inverse: f⁻¹ : B → A No information is lost moving from A to B or back. Preserving Structure Bijectivity alone is not enough. An isomorphism must also preserve structure. For groups, this means: f(a ⋆ b) = f(a) ∘ f(b) for all a, b ∈ A ,...

Proof by Induction: The Sum of Squares Formula Theorem. For any integer n ≥ 1, ∑ i=1 n i² = (1/6)·n·(n+1)·(2n+1) Proof (by mathematical induction) 1. Basis Step For n = 1: LHS = 1² = 1 RHS = (1/6)·1·(1+1)·(2·1+1) = (1/6)·1·2·3 = 1 Therefore, the formula holds for n = 1. 2. Inductive Hypothesis Assume the statement is true for some integer k ≥ 1: ∑(i=1→k) i² = (1/6)·k·(k+1)·(2k+1) 3. Inductive Step We must show it is true for n = k + 1: ∑(i=1→k+1) i² = [∑(i=1→k) i²] + (k+1)² = (1/6)·k·(k+1)·(2k+1) + (k+1)² = (1/6)(k+1)[k(2k+1) + 6(k+1)] = (1/6)(k+1)[2k² + 7k + 6] = (1/6)(k+1)(k+2)(2k+3) This matches the same form with k replaced by k+1: (1/6)·(k+1)·((k+1)+1)·(2(k+1)+1) Hence, the formula holds for n = k + 1. 4. Conclusion Since the statement is true for n = 1 (basis step), and true for n = k + 1 whenever it is true for n = k (inductive step), it follows by the principle of mathematical induction that ∑(i=1→n) i² = (1/6)·n·(n+1)·(2n+1) for al...

The Method of Differences — A Clean Proof of the Sum of Cubes The method of differences is a remarkably elegant tool for evaluating finite sums. When each term of a series can be written in the form f(r+1) − f(r) , the sum “collapses” — all interior terms cancel, leaving only a boundary expression. This behaviour is called a telescoping sum . 1) Telescoping Sums Assume the general term u r can be written as: u r = f(r+1) − f(r). Then the finite sum from r = 1 to r = n becomes: Σ u r = Σ ( f(r+1) − f(r) ). To see what happens, write out a few terms: u₁ = f(2) − f(1) u₂ = f(3) − f(2) u₃ = f(4) − f(3) ⋮ uₙ = f(n+1) − f(n) When these are added, everything cancels except the first and last pieces: Σ u r = f(n+1) − f(1). This is the essence of the method: interior structure disappears, leaving just the difference between the final and initial states. 2) A Classic Application — The Sum of Cubes We will use this technique to prove the well-known ...

Injective, Surjective, and Bijective Functions In mathematics, a function describes how elements of one set are assigned to elements of another. Three important properties capture how completely and uniquely a function connects its domain to its codomain: injective , surjective , and bijective . These properties tell us whether different inputs can share the same output and whether every possible output is used. Injective (One-to-One) A function is injective if different inputs always produce different outputs. No two distinct elements in the domain are allowed to map to the same result in the codomain. Formally, a function f : A → B is injective if: f(a₁) = f(a₂) ⇒ a₁ = a₂ Equivalently, if a₁ ≠ a₂ , then f(a₁) ≠ f(a₂) . Example: f(x) = e x from ℝ → ℝ is injective. Different inputs produce different outputs, but not every real number is reached, so it is not surjective. Surjective (Onto) A function is surjective if every element of the codomain is reach...

What Is a Group in Mathematics? In abstract algebra, a group is a set G together with a binary operation (written as * ). The pair (G, *) is called a group when the operation satisfies the four conditions below. Closure: for all g₁, g₂ ∈ G , the product g₁ * g₂ is still in G . Identity: there exists an element e ∈ G such that for all g ∈ G , e * g = g * e = g . This element e is called the identity. Inverses: for each g ∈ G there exists an element g⁻¹ ∈ G such that g * g⁻¹ = g⁻¹ * g = e . Associativity: for all g₁, g₂, g₃ ∈ G , g₁ * (g₂ * g₃) = (g₁ * g₂) * g₃ . These four conditions are exactly what is required for (G, *) to be a group. Side note: Commutativity An operation is commutative if swapping the elements does not change the result: a * b = b * a . Commutativity is not required for a group to exist. It is important not to confuse commutativity with associativity . These are distinct i...

Essential Elements of Algebra Problem Solving Solving equations in algebra is about preserving truth while working step by step toward the unknown. Although equations can look complicated, most of the time we are simply applying a small set of rules with care. If these rules are followed, the solution you reach will be valid; if they are broken, the result becomes unreliable. This post introduces three essential principles that underpin almost all algebraic manipulation. Follow them consistently, and you will have a strong foundation for solving equations with confidence. 1) Start with a True Statement Everything in algebra begins with a statement that is already true. If the starting point is false, no amount of manipulation can produce a trustworthy conclusion. When you write an equation such as: 2x + 3 = 13 you are asserting that doubling a number and then adding three gives thirteen. This is your initial truth. All further steps must preserve this truth. If y...

Function Composition: A Simple Way to Organise Your Mathematics Function composition is one of the most useful tools in mathematics. It allows us to combine several steps into a single process, keeping our work neat, organised, and easy to reuse. Rather than performing one operation after another by hand, composition lets us build those steps into a single function. Once you become comfortable with function composition, you never want to go back to doing everything manually. It reduces clutter, helps you work systematically, and allows you to achieve remarkable results with only a few lines of equations. What Is Function Composition? A function takes an input, performs an operation, and produces an output. Function composition takes this idea further: it links functions together so that the output of one becomes the input of the next. We write this as: (f ∘ g)(x) = f(g(x)) This means that g acts first, then f . The circle symbol ∘ simply means “do one funct...

Below are the workings you can use to find the intersection points of two quadratic equations . Part 1 Part 2   There is a free interactive graph related to these workings here: https://www.desmos.com/calculator/0twcp4hwx2

Today I was watching a video about how to construct a dodecahedron . In the first part of the clip, it was shown how to get the golden ratio with a specific type of geometric construction . I was absolutely intrigued by it and decided to figure out if the construction did indeed yield the result phi. Remarkably and beautifully, I was able to confirm the result. My workings are shown below. I live for these eureka moments! Part 1 Part 2 Part 3   Part 4

Below I demonstrate how to derive tan(π/8) , sin(π/8) and cos(π/8) from scratch. These workings are a bit similar to the workings for tan((3π)/8) , sin((3π)/8) and cos((3π)/8) which can be found here : https://geometricbites.blogspot.com/2021/08/how-to-derive-tan38-sin38-and-cos38.html The edge with length R √(2- √(2)) was actually derived in the tan((3π)/8), sin((3π)/8) and cos((3π)/8) post, so doesn't need to be found again. Part 1  Part 2  Part 3  Part 4

Below I demonstrate how to find the line that passes through two intersecting circles. Part 1 Part 2 Part 3   There is a free interactive graph that shows why this is correct. https://www.desmos.com/calculator/z9rh2ri9bz

In this post I demonstrate how to find out where a line and circle intersect on a 2 dimensional plane. The first step is to write out the equations of a line and a circle. We then make sure for both equations 'y' is isolated . Once we have the equations for a line and a circle whereby 'y' is isolated, we can then go about finding the values of 'x' for where the line intersects the circle. Like so... After we have found the values of 'x' using the quadratic formula , we then plug them back in to y=mx+c to get the values of 'y' for which the line intersects the circle. And that's it basically. Below is a free interactive Desmos graph related to this work: https://www.desmos.com/calculator/w3tsnjajtx

Below are the workings required to build an interactive graph that finds the points where a line and quadratic equation intersect . Here 'a' is not equal to 0 as it becomes part of the denominator a fraction . Also, if 'a' were equal to 0, we'd have a line intersecting a line, not a parabola (for instance). When we have the formula for 'x', we just plug it back into the linear equation to get the outcomes for 'y'. And there we have it, the points where a line and quadratic equation intersect. To see the interactive graph I was talking about, visit the Desmos link below: https://www.desmos.com/calculator/sqzjajbhea