Update index.html

Added content, fixed schemantics

Author: Gmac4247

index.html

@@ -187,7 +187,7 @@
 },
 "dateCreated": "2019-01-11",
 "datePublished": "2020-01-11",
-"dateModified": "2025-11-14",
+"dateModified": "2025-11-15",
 "description": "Introducing the best-established and most accurate framework to calculate area and volume.",
 "disambiguatingDescription": "Exact, empirically grounded and rigorously proven formulas over the conventional approximations.",
 "headline": "Introducing the Core Geometric System ™",
@@ -527,6 +527,8 @@
 <br>
 <br>
 <b>5. Geometry</b>
+</p>
+</div>
 <br>
 <br>
 <br>
@@ -660,6 +662,19 @@ <h2 style="font-size:160%;margin:7px">Mathematical operations</h2>
 <h3 style="font-size:160%;margin:7px">Fractions</h3>
 <br>
 <br>
+<p style="margin:12px;">Fractions are non-whole numbers.
+<br>
+E.g. the results of breaking a branch in half are 2 half branches.
+<br>
+A half is mathematically expressed as 1 / 2, or 0.5.
+<br>
+Halving a half gives two quarter pieces... ( 1 / 2 ) / 2 = 1 / 4 = 0.25, and so on.
+<br>
+The numerals after the decimal point are called decimals.
+<br>
+I.e. 1 / 8 = 0.125. The decimals are 1 tenth ( 1 / 10 ), 2 hundredths ( 2 / 100 ) and 5 thousandths ( 5 / 1000 ).
+<br>
+<br>
 <section>
 <strong style="margin:12px;">➕ Addition</strong>
 <br>
@@ -796,13 +811,13 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 Raising a number to a power means multiplying it by itself.
 <br>
 <br>
-Example #1: 2³ = 2^3 = 2 × 2 × 2 = 8
+Example #1: Raising to the 3rd power means multiplying the number by itself 3 times. 2³ = 2^3 = 2 × 2 × 2 = 8
 <br>
 <br>
-Example #2: 3^(-4) = 1 / 3⁴ = 1 / ( 3 × 3 × 3 × 3 ) = 1 / 81
+Example #2: Raising to the - 4th power means the reciprocal of the 4th power. 3^(-4) = 1 / 3⁴ = 1 / ( 3 × 3 × 3 × 3 ) = 1 / 81
 <br>
 <br>
-Example #3: 4^(1/2) = √4 = 2
+Example #3: Raising to a fraction gives a root. 4^(1/2) = √4 = 2
 </p>
 </section>
 <br>
@@ -1012,7 +1027,6 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
     <img class="center-fit" src="trigonometry.png" alt="figure-Trigonometry" id="trigonometry">
     </div>
 <br>
-<div>
 <p style="margin:12px;">
 In a right triangle:
 <br>
@@ -1101,7 +1115,7 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 </mrow>
 </math>
 </p>
-</div>
+</section>
 <br>
 <br>
 <br>
@@ -1633,7 +1647,7 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 
 </script>
 </div>
-<div>
+<section>
  <p style="margin:12px;">The area of a triangle equals exactly the half of the area of a rectangle with a width equal to the base of the triangle and length equal to the height of the triangle.
 <br>
 <br>
@@ -1771,7 +1785,6 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 </msqrt>
 </mrow>
 </math>
-</div>
 </section>
 <br>
 <br>
@@ -1943,7 +1956,8 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 <br>
 <br>
 <br>
-<section><strong>The Core Geometric System ™ fundamentally shifts its axioms from the abstract, zero-dimensional point to the square and the cube as the primary, physically-relevant units for measurement.
+<section>
+<strong>The Core Geometric System ™ fundamentally shifts its axioms from the abstract, zero-dimensional point to the square and the cube as the primary, physically-relevant units for measurement.
 <br>
 We define the properties of shapes like the circle and sphere not through abstract limits, but through their direct, rational relationship to these foundational units, resulting in the use of the rational constant 3.2 instead of the irrational pi.
 </strong>
@@ -1997,10 +2011,11 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 <br>
 <br>
 <section>
-<strong style="font-size:160%;margin:7px">Area of a circle</strong>
+<strong style="font-size:160%;margin:7px">Area and circumference of a circle</strong>
 <br>
 <br>
-For centuries, the circle has been a symbol of mathematical elegance—and π its most iconic constant. But beneath the surface of tradition lies a deeper question: Are the formulas we use truly derived from geometric logic, or are they inherited approximations dressed in symbolic authority?
+<section>
+<p style="margin:12px;">For centuries, the circle has been a symbol of mathematical elegance—and π its most iconic constant. But beneath the surface of tradition lies a deeper question: Are the formulas we use truly derived from geometric logic, or are they inherited approximations dressed in symbolic authority?
 <br>
 <br>
 This article revisits the foundations of circle geometry, challenging long-held assumptions and offering a more exact, algebraic alternative.
@@ -2063,9 +2078,9 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 <br>
 Similarly, the area formula A = πr² is not a direct result of calculus. It’s reverse-engineered by multiplying the circumference formula C = 2πr by half the radius—treating the area as the sum of infinitesimal rings. While the method is algebraically valid, it bypasses the geometric logic that defines area: the comparison to a square.
 </p>
+</section>
 <br>
 <br>
-</section>
 <br>
 <br>
 <section>
@@ -2193,6 +2208,7 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 These values are exact, rational, and logically derived. They can be verified numerically, but more importantly, they can be proven algebraically—without relying on infinite fractions, symbolic shortcuts, or flawed assumptions.
 </p>
 </section>
+</section>
 <br>
 <br>
 <br>
@@ -2380,7 +2396,6 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 </script>
 <p style="margin:12px;" id="circle-area"></p>
 </div>
-</section>
 <br>
 <br>
 <section>
@@ -2756,10 +2771,11 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 The quadrant method proves that the area of a circle equals exactly 3.2 × radius², thus ruling out the validity of the π.
 </p>
 </section>
+</section>
 <br>
 <br>
 <br>
-<div>
+<section>
     <p style="font-size:160%;font-weight:bold;margin:7px">Area of a circle segment</p>
 <br>
 <div>
@@ -2879,7 +2895,7 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
     <mi>r</mi>
   </mrow>
 </math>
-</div>
+</section>
 <br>
 <br>
 <br>
@@ -3036,6 +3052,7 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 </div>
 <br>
 <br>
+<section>
 <strong>Proof</strong>
 <br>
 <br>
@@ -3230,7 +3247,6 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 </section>
 <br>
 <br>
-<section>
 <p style="margin:12px;"><b>Conclusion: Time to Move On</b>
 <br>
 <br>
@@ -3247,7 +3263,6 @@ <h4 style="font-size:160%;margin:7px">Powers, numeral systems and basic geometry
 </p>
 </section>
 </section>
-</div>
 <br>
 <br>
 <br>