Children

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

Pythagorean Suits

Pythagorean suits are important in architectural constructions, which aroused the interests of the Greeks. They were extremely important, thanks to them we have one of the most beautiful and lasting architectures, which survives to this day. The Pythagorean suits Initially, let’s define what the Pythagorean suit is. Definition (Pythagorean Terno): is any set of natural numbers $x,y$ and $z$ (terns) such that: $$x^2 + y^2 = z^2$$ This definition was used by Euclid in Alexandria, who used concepts from the Pythagorean theorem, which corresponds to the right triangle:

October 21, 2022 Read

Ranges in real numbers

Open and closed doors on the straight Include and exclude endpoints. How do interval and inequality symbols change? Explore the controls and explain what changed in your own words. Ative o JavaScript para ver esta figura interativa. The intervals will be represented as a subset of the real numbers or better known as the straight line. These subsets are infinite, because no matter how many smaller intervals you take, they will still be infinite. Unlike the set of natural numbers or integers.

October 19, 2022 Read

Inequalities of real numbers

So far, we have only worked with equalities, in equations we work with scales, which is the notion of balance. In this post we will work with the opposite, that is, imbalance. Difference between equality and inequality To illustrate the notion of equality and inequality, look at the following sentences: Letícia is 2 years older than Alexandre (equality); Letícia is at least 2 years older than Alexandre (inequality). In this case, we have different meanings. Let $L$ be Letícia and $A$ be Alexandre, we can write the sentences as:

October 17, 2022 Read

Notion of Infinity

Infinity is an old discussion, around the 5th century BC with Zeno’s paradox. It was only during the Renaissance around the 15th century AD that this concept reappeared and in this post, we will talk about it. One of the applications is the law of never dividing by zero. The Infinity The concept of infinity was first studied by some famous mathematicians, such as: Marin Mersenne, friend of Descartes; Adrien-Marie Legendre; *James Joyce; John Wallis, creator of the infinity symbol ($\infty$).

October 17, 2022 Read

2° degree equation

In the previous post, the technique used by Bhãskara to solve 2th degree equations was shown. Many mathematicians between the 15th and 17th centuries developed different forms for 2th degree equations. One of them was François Viète (1540-1603). Proposal by François Viète François Viète, inspired by Bhãskara, used the shared scarf method, which we will demonstrate in the next topic. François Viète Excerpt from some of the topics studied by François Viète.

October 15, 2022 Read

Compound interest and Euler's number

John Napier was so important for applications in finance that he motivated Leonhard Euler to venture into this area. One of the irrational numbers most important with application in finance, is due to these two mathematicians. Compound Interest The history of compound interest and the number $e$ was thanks to Jacques Bernoulli between (1654-1705), who became interested in discovering the maximum value of interest on loans using the compound interest technique.

October 15, 2022 Read

Bhaskara

One of the biggest traumas for school children in modern times is the second-degree equation, where they make them memorize formulas. This memorize make students hate math. Notice what it is like to teach in a modern school. The equation of 2° degree is given by: $$ax^2 + bx + c = 0$$ in which $$\Delta = b^2 - 4ac$$From this, we obtain that the roots of Bhãskara’s equation: $$x = \frac{-b\pm \sqrt{\Delta}}{2a}$$ Bhãskara’s work The work of Bhãskara(around 1070 AD), which cites the writings of Śridhara (around 1030 AD), is what went down in history as the author of solving the equations of 2° degree. The first to work on this topic was Al-Khowarizm, in approximately 1000 AD, who is cited by Śridhara.

October 14, 2022 Read

Simple Interest

The Sumerians were the first to use concepts of contracts, invoices, receipts, and rudimentary credits. They started the concept of finance after the Babylonians around 2000 BC (Hammurabi’s code), with the concept of interest that was paid with seeds or other goods. The emergence of coins was around the 7th century BC. C. in Lydia, present-day Türkiye. Over time, the accumulation of these coins began to be a problem, as they could lose control of how much they had saved. The concept of a bank began to emerge with this problem.

October 14, 2022 Read

Increases and Discounts

Since childhood, the notion of finances is important, even if simple. The starting point is to learn about percentage increases and discounts. This concept can be applied to the price of goods, among others. Correction factor The beginning of the concept of increase and discount is given by a correction factor, applied to the price or to something that is increased or decreased. Let $p$ be the percentage applied to the price of the merchandise. Thus, the correction factor $f_c$ is given by:

October 10, 2022 Read

Equiprobable Events

In this post we will talk about equiprobable events. This concept is what we cover in teaching children and is the starting point for the concept of probability. What is an equiprobable event? Equiprobable events are events with the same probability of occurring, that is, they have the same chance of occurring. It appears a lot in probability theory. This way, we will define it as: Definition (Equiprvable Events): Let there be a sample space $\Omega$ and a random experiment from this space. When we take a class of events from this finite space $\mathcal{A}=\\{e_1, e_2,\dots,e_n\\}$. Events are equiprobable if these events have the same probability of occurring. $$\mathbb{P}(e_1) = \mathbb{P}(e_1) = \dots = \mathbb{P}(e_n) =\dfrac{\text{número de eventos favoráveis} }{\text{total de casos} }$$

October 8, 2022 Read

Introduction to Real Numbers

The first model for real numbers was proposed by the Greeks, from Pythagoras to Euclid. The beginning of everything came with irrational numbers, like $\sqrt{2}$ and $\pi$. Evolution of number concepts In this book for children we begin the construction of the entire sequence until the introduction to real numbers. We start with concepts of counting and measuring with natural numbers, which is the starting point for building the number line.

October 3, 2022 Read