Mathematics for Children

Triangles and Circles

The triangle is one of the most frequently discussed geometric figures in the study of geometry. Its relationship with circles is fundamental to understanding various geometry results. Some of these results are called notable points of the triangle. Similarity of Triangles In the proportions post, we saw concepts of similarity of triangles. Here we will show that the similarity of triangles can be demonstrated by circles. Triangles inside the circle Lemma 1: Let AB and CD be different chords of a circle that intersect at a point P inside the circle. Then $$\dfrac{AP}{CP} = \dfrac{PD}{PB} $$

October 1, 2022 Read

Angles and Circumferences

Learning about angles in circles is fundamental. This concept was extensively addressed by Euclid and the Greeks in general, such as Pythagoras and Archimedes. Bow Definition: Let $A$ and $B$ be points on a circle in a plane. The straight line $\vec{AB}$ (arc) separates the plane into two half-planes, each containing a part of the circle. Central Angle Definition: Let $A$ and $B$ be points of a circle on a center plane $C$. The central angle $\angle ACB$ determined by $A$ and $B$ or determined by the chord $AB$.

September 30, 2022 Read

Fundamentals

In this post we will delve deeper into some probability concepts. In previous posts we talked about the concepts of functions, which according to Descartes is the act of saying what one hears. Given concepts of events and sample spaces, we will calculate the chances of these events occurring with the concept of a function, in this case probability. Fundamentals In previous posts, we talked about the concept of function. In this way, René Discards used an ear as a symbol.

September 29, 2022 Read

Introduction to Probability

In this post we are going to talk a little about probability, for this we must bear in mind that working with this theory it is essential to evaluate: human intuition phenomena of nature (physis or physics) among others Over the years, mathematics has also been based on these concepts. Years 0 to 1637 - Drawing nature; Years 1637 to 1830 - Decode nature; Years 1830 to 1890 - Imitate and engineer nature; Years 1890 to 1980 - Control nature; Years 1980 to today - Creating nature. Initial notions of Probability Probability is based on human learning principles. For this, we use experiment concepts.

September 28, 2022 Read

Circumferences

The main work on geometry, Euclid’s book Elements, has several results for straight lines and circles. In this post, we will show several results for this subject. Basic Settings Below we define some fundamental terms for working with circles. Center Definition (Center): is a point that is the same distance from any point on the circle. Ray Definition (Radius): is any straight line segment that runs from the center to the circumference.

September 27, 2022 Read

Properties of Logarithms

From the semiotically active logarithm notation (abbreviated Mirabilis) we obtain the properties that are used in current times. Next, we present the properties. Law of Multiplication (Multiplicatio Canon) $$ \begin{aligned} \displaystyle \log_{2^6} 64 &= \\ &= \log_{2^6} 4~.~16 = \log_{2^2} 4 + \log_{2^4} 16 \\ &= \log_{2^6} 8~.~8 = \log_{2^3} 4 + \log_{2^3} 16 \\ &= \log_{2^6} 2~.~32 = \log_{2^1} 2 + \log_{2^5} 32 \\ \end{aligned} $$ $$\log_{C} A~.~B = \log_{C} A + \log_{C} B$$ Law of division (Tangentiplus Canon) $$ \begin{aligned} \log_{2^4} \frac{64}{4} &= \log_{2^6} 64 - \log_{2^2} 4 \\ \log_{5^1} \frac{125}{25} &= \log_{5^3} 125 - \log_{5^2} 25 \end{aligned} $$ $$\log_{C} \frac{A}{B} = \log_{C} A - \log_{C} B$$ Law of Power (Potestati Canon) $$ \begin{aligned} \log_{2^6} 4^3 &= \log_{2^6} 64\\ &= \log_{2^6} 4 ~.~ 4 ~.~ 4\\ &\overset{\text{Multiplicatio Canon}}{=} \log_{2^2} 4 + \log_{2^2} 4 + \log_{2^2} 4\\ &= 3~.~\log_{2^2} 4\\ \end{aligned} $$ $$\log_{C} A^B = B~.~\log_{C} A$$ Application for Planck constant To understand the power of this notation, let’s calculate Planck’s constant.

September 26, 2022 Read

Circle Area

Approximately 2300 years ago Eudoxus, Archimedes and others encouraged the area of the circle through the exhaustion method, which consists of circumscribing and inscribing regular polygons in the figure that is intended to evaluate the area. Area of the circle - Greek Method Initially, let’s define the area of a square region. Definition: The area of a square region of side $a$ is $a^2$. The Greeks verified that every polygonal region can be inscribed in a square through a finite number of operations with a ruler and compass.

September 25, 2022 Read

Introduction to Logarithm Table

A tool that served as a calculator in the Middle Ages, today serves as a horror story for children. The logarithm is one of the most famous works, so much so that it is treated as Mirabilis Abreviatio, that is, a semiotically active language. In this post we need concepts of potentiation seen in other topics covered. If you have questions, check the other posts. John Napier - the Baron of Merchiston John Napier - the Baron of Merchiston This work is due to John Napier, the Baron of Merchiston, in his work Mirifici Logarithmorum Canonis Descriptio (1614), which can be translated as A Wonderful Description of the Evolution of the Laws of God. To escape church sanctions, the work was again translated into A wonderful description of the laws of evolution of numbers. Years later Leonhard Euler honored Naiper with the Naperian logarithm. However, the irrational number $e$ was named after Napier, but $e$ comes from Euler.

September 25, 2022 Read

Pi Number

The secret of the wheels Change the diameter and compare C/d. The contour changes, but the ratio in the model remains π. Explore the controls and explain what changed in your own words. Ative o JavaScript para ver esta figura interativa. Archimedes’ constant, known as pi (Π) was widely studied in Greece (around the 3rd century BC). It all started with the study of one of the most important figures in geometry, which is the circle. This constant is obtained by the ratio ($\div$) of the length of the circumference and the diameter.

September 20, 2022 Read

Introduction to Irrational Numbers

In this group of posts the objective is to present non-rational numbers, which we call irrational. For this it is important to talk about the Pythagorean constant $\sqrt{2}$. Until the emergence of this constant, the Pythagorean school followed in the footsteps of the Sumerians, Plato’s academy, Alexandria’s school, among others. Historical definition of the Square Root In introductory posts, we saw concepts of square root, which is nothing more than the side of the square.

September 19, 2022 Read

Graphs for integers

A function graph with integer inputs Compare this dot plot with the bar graphs in class. Here, each pair represents an input x and an output f(x); In a bar chart, the horizontal axis can also represent categories. Em telas pequenas, deslize o gráfico na horizontal para ler os detalhes. Change the rule and look at the table. We do not connect the dots when the domain is ℤ. Ative o JavaScript para ver esta figura interativa.

September 16, 2022 Read

Classification in mathematics

Classification is a very important topic, used since the Greeks with Archimedes. In biology, for example, classification is done by groupings, which organize living beings into categories such as kingdom, phylum, class, order, family, genus and species. In mathematics, this first contact with classification is the grouping of numbers into sets. Next, let’s talk about classification in terms of groups and categories. Groups In this section we will talk about groups in terms of classification and not group theory, which is an abstract theory and will be covered in pure mathematics or history of mathematics. This theory is not for children yet.

September 15, 2022 Read