Mathematics for Children

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

Equivalence between Sets

In previous posts we talked about how to represent, concepts of relevance and presented some important sets. Here, we are going to talk about equivalence and equality between two sets. Equality of a set Initially, let’s define the equality of two sets. Definition (Equality of two sets): Two sets are equal if they have exactly the same elements. Example 1: Let the sets $A$ and $B$ have the same elements:

November 12, 2022 Read

Introduction to Sets

At this point in the posts, we will talk about sets as a collection of elements, which can be abstract or not. The abstract ones are those that are in the mathematical context, such as a set of odd integers, among others. The non-abstract ones are in everyday life, such as provinces in Japan, students at USP, among others. Representation of a set The representation of sets are given by capital letters such as:

November 11, 2022 Read

Introduction to Statistics

Statistics is one of the most applied areas of mathematics, with greater use in everyday life. For centuries, it was used unconsciously by people as a merely descriptive character and a record of occurrences. The first uses The first activities were around 2000 BC and were used in the census of Chinese agricultural populations. At the beginning of the 19th century, great mathematicians entered the scene, for example, the Frenchman Simon Laplace and the German Carl Friedrich Gauss (1777 –1855). The latter arises with applications of the “normal distribution” for modeling measurement errors.

November 11, 2022 Read

Interval Operations

The intervals of the straight line or simply straight line remind us of the concept of continuity (continuous numbers). With natural numbers, for example, we cannot obtain this continuity, which we call discrete numbers. Be careful, there are different types of infinity. In this post, we will cover operations with ranges of real numbers. Ranges in discrete numbers Natural, integer or rational numbers are said to be countable or enumerable sets. When an interval is finite, we can also classify it into discrete intervals.

October 23, 2022 Read