Children

Arithmetic Progression

From Gauss's story to the justification of the formula The school narrative associated with Gauss highlights the grouping of the ends of a sum. The idea is valid beyond that story: we can demonstrate it for any finite arithmetic progression. Your turn to investigate: Write 2, 5, 8, 11, 14 and, below, the same sequence in reverse. Add the columns. Why is the result for each column the same? Discover the historical source on MacTutor, from the University of St Andrews.

April 10, 2025 Read

Nth Root

Arithmetic enzyme root Definition Given a real number $a \geq 0$ and an integer $n \geq 2$, there exists a unique real number $b \geq 0$ such that $b^n = a$. The number b we will call the arithmetic enzymatic root of a and we will indicate it by the symbol $\sqrt[n]{a}$ where a is called the radicand and n is the index. Examples $\sqrt[5]{32} = 2$ because $2^5 = 32$ $\sqrt[3]{8} = 2$ because $2^3 = 8$ $\sqrt{9} = 3$ because $3^2 = 9$ $\sqrt[7]{0} = 0$ because $0^7 = 0$ $\sqrt[6]{1} = 1$ because $1^6 = 1$ Observations

March 27, 2025 Read

Domain and Image

Domain and image Definition: Let R be a relation of A to B. The set D of all the first elements of the ordered pairs belonging to R is called the domain of R. x The set Im of all the second elements of the ordered pairs belonging to R is called the image of R. y It follows from the definition that DCA, and ImCB. Examples: If A = {0,2,3,4} and B = {1,2,3,4,5,6} what is the domain and the image of the relation R = {(x,y) E A X B y is a multiple of x}? Using the arrow scheme, it is easy to see that D is the set of elements of A from which arrows depart and that Im is the set of elements of B to which arrows arrive, meaning: R = {(2.2), (2,4), (2,6, (3,3), (3,6) (4,4),)} D = {2,3,4} Im = {2,3,4,6}

March 21, 2025 Read

Circumference ratio in regular polygons

In previous posts we talked about Stewart’s theorem, here we will talk about the circumference relationship in regular polygons. The relationship between circles and regular polygons is a fascinating and complex topic in geometry. Regular polygons are geometric figures with congruent sides and angles, and the circle can be seen as a limiting case of a regular polygon with an infinite number of sides. When a regular polygon is inscribed in a circle, each of its vertices touches the circle, and the center of the polygon coincides with the center of the circle. This allows precise relationships to be established between the radius of the circle and the dimensions of the polygon, such as the side and apothem (the distance from the center to a side).

July 18, 2024 Read

Stewart Relationship

In previous posts we talked about ptheorems for any triangles. The Stewart Relation, also known as Stewart’s Theorem, is an important formula in geometry that relates the lengths of the sides of a triangle to the length of a cevian. A cevian is a line segment that starts from a vertex of a triangle and meets the opposite side. This relationship is useful for solving problems involving determining the lengths of segments within a triangle, such as medians, bisectors, and heights.

July 12, 2024 Read

Theorem for any triangles

In previous posts we talked about point power. The Sine Theorem is a fundamental mathematical tool in trigonometry, which establishes an important relationship between the sides of a triangle and the sines of its angles. This theorem is applicable to any type of triangle, be it acute, obtuse or scalene, and is essential for solving problems involving the geometry of triangles. The Law of Sines states that, in any triangle, the ratio between the length of one side and the sine of the opposite angle is constant and equal for all three sides and angles of the triangle.

July 12, 2024 Read

Pythagoras Theorem

The Pythagorean Theorem is one of the best-known and fundamental propositions in mathematics, establishing an elegant relationship between the sides of a right-angled triangle. The history of this theorem is as fascinating as its mathematical application, as it spans cultures and eras, revealing how mathematical knowledge is accumulated and shared across generations. Traditionally attributed to the Greek philosopher and mathematician Pythagoras, who lived between approximately 570 BC and 495 BC, the theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, known as legs.

July 11, 2024 Read

Right triangle relations

The right triangle is a geometric figure that fascinates mathematicians and students due to its simultaneous simplicity and complexity. The metric relations of the right triangle are a set of formulas that relate the lengths of the sides and the projections of the legs on the hypotenuse. These relationships are fundamental to understanding more advanced concepts in trigonometry and analytical geometry. The best known of these relationships is the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, the legs. Mathematically, this is expressed as $a^2 = b^2 + c^2$, where $a$ is the hypotenuse and $b$ and $c$ are the legs.

July 11, 2024 Read

Exercises for Adding Natural Numbers

Warm-up with figures Solve first with the figures, then write the calculation on paper. Solved Exercises on Adding Natural Numbers - Elementary School Example 1: Simple sum Statement: Maria has 3 candy and John has 4 candy. How many bullets do they have together? Resolution: Maria has 3 candy and John has 4 candy. To find out how many bullets they have together, we add:

July 10, 2024 Read

Introduction to Arithmetic Progression

A school story and an idea to try A much-told story about Gauss presents him, as a boy, quickly solving a sum by grouping terms. Let’s use this narrative as an invitation to investigate patterns. Your turn to investigate: Write 1 + 2 + … + 10. Join the first term to the last, the second to the penultimate and continue. How many pairs can you get? What is the sum of each pair? Discover the historical source on MacTutor, from the University of St Andrews.

July 10, 2024 Read

Point Power

In previous posts we talked about inscribed angle. The power of a point is a real number that reflects the relative distance of a given point to a given circle. To understand this better, imagine you have a point A outside a circle. Draw two secants of A that intersect the circle at two points each. The power of point A is then defined as the product of the distances from A to one point of intersection with the product of the distances from A to the other point of intersection on the other secant.

July 10, 2024 Read

Segment angle

In previous posts we talked about inscribed angle. The segment angle is defined as the angle formed between a chord of a circle and the tangent to the circle at one of the ends of the chord. This angle can be seen as a special case of the inscribed angle, where the vertex of the angle is precisely on one of the points that define the arc. To understand better, imagine a circle with a rope extended from one point to another. At one of the points where the rope touches the circle, draw a tangent line. The angle formed between this tangent and the chord is what we call the segment angle. An interesting property of this angle is that it is half the measure of the central angle that subtends the same arc.

July 9, 2024 Read