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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

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