Children

Straight lines

In this post we are going to talk about an important concept for plane geometry, which are concepts of lines, line segments, semi-lines, among others. Line segment Definition (Line segment): given two distinct points, the union of the set of these two points with the set of points that are between them, is a line segment. In symbolic terms we have the sets: $$\overline{AB} = \\{A,B\\}\cup \\{\text{for all } X \text{ between } A \text{ and } B\\}$$Consecutive segments Definition (Consecutive Segments): two line segments are consecutive if and only if one endpoint of one of them is also the endpoint of the other.

December 2, 2022 Read

Primitive Notions

Here we start the series of posts for plane geometry. In it, we will talk from very primitive concepts, to more advanced concepts, aimed at children (elementary to medium). Start using geometry The Greeks of ancient civilization were one of the most famous people in the art of thinking. They were responsible for the heyday of philosophical, scientific, political, social and artistic ideals and had a decisive influence on the western world.

December 1, 2022 Read

First Degree Functions

The beginning of learning about functions goes through first degree functions. In this post, we will present some first-degree functions. Among them are constant, linear and related. Constant function The constant function is a plateau, something flat or simply a horizontal line. Below we present the formal definition. Definition (Constant Function): an application $f$ of $\mathbb{R}$ in $\mathbb{R}$ is called a constant function when each element $x \in\mathbb{R}$ always associates the same element $c \in\mathbb{R}$.

November 28, 2022 Read

Dispersion Measures

Dispersion measures are fundamental, because unlike mathematics, statistics work with random variables. For example, the big media worked with deterministic values, such as candidate A has 45% and B with 30%. For mathematics, A is clearly ahead, but for statistics, I don’t know. Today they present it in a statistical way, as they talk about the “margin of error”. If the “margin of error” is 15%, the two are technically tied.

November 27, 2022 Read

Position Measurements

Descriptive statistics are numbers that summarize and describe all information. In this case, the data sets or simply data. But. This information will depend on what we evaluate, such as population or sample, as stated in the previous post. Getting Started The central theme of this post is just “describing” the data. Therefore, they do not represent more specific conclusions of the sample in relation to the population. The technique used to extend conclusions from the sample to the population is statistical inference, a topic that does not make sense to children. An undergraduate statistics course is the most appropriate place.

November 26, 2022 Read

Basic Concepts of Statistics

Mathematics works with deterministic values, this means that if x is equal to 10, it is 10 and that’s it. Statistics work with random variables. For example, if you measure the same item several days in a row, the measurement will not be the same. Many factors can influence daily measurement, such as your body temperature, environment, measuring instrument, among others. Population and Sample Concept In statistics, some concepts are fundamental to understanding what this science is, which is widely used in everyday life. Below we present two of them:

November 25, 2022 Read

Function Concept

The law of correspondence is the modern definition for the subject of functions. In addition, it is important to define concepts of domain and image, already explained in the function introduction post, under the vision of René Descartes. Tribute to the Bernoulli family No family in the history of mathematics has produced as many famous mathematicians as the Bernoulli family. Originally from the Spanish Netherlands, this family emigrated in 1583 to Basel, Switzerland, fleeing the war. About a dozen members of the family achieved renown in Mathematics and Physics, four of them being elected as members foreigners from the French Academy of Sciences.

November 22, 2022 Read

Correspondence law

The law of correspondence is the modern definition for the subject of functions. In addition, it is important to define concepts of domain and image, already explained in the function introduction post, under the vision of René Descartes. Tribute to Abel Niels Henrik Abel, from a large and poor family, was the son of the shepherd from the small village of Findo, in Norway. At the age of 17, his teacher insisted that he read the great mathematical works, including Gauss’s “Disquisitiones” (Research). At this time, Abel managed to generalize the binomial theorem that Euler had only proven for rational powers.

November 21, 2022 Read

Understanding Relationships

The concept of relations is the starting point for the concept of functions in real numbers. In addition, concepts of ordered pair, Cartesian product and the Cartesian plane itself. One of the main mathematicians who contributed to this concept was Julius Wilhelm Richar Dedekind. Tribute to Dedekind Our honoree today is Julius Wilhelm Richar Dedekind. He was one of four children in a Lutheran family from Braunschweig, Germany. At the age of 19 he entered his doctorate in Gottingen, at 22 he completed it. A student of Dirichlet, he taught secondary education in Brunswick until the last years of his life. He was very concerned about the nature of functions and numbers. In this way, he has been focused on the problem of irrational numbers since 1858 when he taught Calculus.

November 21, 2022 Read

Introduction to Function Theory

Functions are one of the most important subjects in mathematics. René Descartes, a pioneer in the study of the Cartesian plane, defined it as the act of listening and speaking. As a result, the first uses of the function were represented with parentheses, precisely because of the ear. Discourse on the Method (Discours de la Methode) How are we going to start the speech? According to Leibniz and Euler:

November 20, 2022 Read

Diagram Operations

Venn Euler diagrams are useful for a visual representation of sets. In OBMEP 2022 a question appeared, using this theory. In this post, we will describe operations with these diagrams. Union of sets Union of sets is one of the most used operations on sets. It will be the starting point for operations with sets, defined by: Definition (Union of sets) Given two sets $A$ and $B,$ the union of these sets is formed by all the elements that belong to $A$ or $B$ ($A\cup B$). In algebraic terms, we can write: $$A\cup B = \\{x;~x\in A ~\text{ ou } x\in B\\}$$

November 14, 2022 Read

Venn-Euler Diagrams

In previous posts we talked about how to represent a set algebraically. In this post, we will show it visually through the Venn Euler Diagram. Venn-Euler Diagram Representation In this type of diagram representation, we use a circle or rectangle, which will depend on the case. For example, the sets: Example 1: Let the set $A = \\{\text{províncias do Japão}\\}$ be represented in a Venn-Euler Diagram: Example 2: Let the set $B = \\{3,4,5,6\\}$ be represented in a Venn-Euler Diagram:

November 13, 2022 Read