מבוא לתורת הקבוצות - 1 - ממה מורכבת המתמטיקה?

Summary

This introductory math course aims to demystify the subject by starting from the basics. It explores the concept of numbers, including natural numbers, zero, negative numbers, rational numbers, and real numbers, with a brief mention of complex numbers. The course will delve into mathematical operations, functions, and the organization of numbers, such as using longitude and latitude to describe locations. It also touches on the idea of sets as a fundamental building block in mathematics, setting the stage for a deeper understanding of mathematical concepts.

Takeaways

• 📚 The course is focused on basic math and aims to explain concepts from scratch.
• 🔍 Math is derived from a Greek word meaning 'knowledge', but its modern definition is more complex and involves dealing with amount, structure, space, and change.
• 🔢 Numbers are a fundamental aspect of math and include natural numbers, zero, negative numbers, rational numbers, real numbers, and more exotic numbers like pi and the golden ratio.
• 🤔 The script challenges the general understanding of math as just dealing with numbers, suggesting that there is more to it.
• 🧩 Numbers can be manipulated through basic operations like addition, subtraction, multiplication, and division, which are considered functions in math.
• 📏 Comparisons, such as determining that Everest is higher than the Dead Sea, involve comparing numbers and describing relations between them.
• 🌐 Every point on Earth is represented by a pair of numbers (longitude and latitude), highlighting the importance of ordered pairs in math.
• 🔗 Ordered pairs and relations are interrelated, with ordered pairs helping to build relations, and functions being a special type of relation.
• 💡 The script introduces the concept that the most basic mathematical object is a set, which will be used to construct other basic objects in the Set Theory course.
• 🏗️ The course will start with the concept of a set and use it to build numbers, ordered pairs, relations, and functions, illustrating the foundational role of sets in mathematics.

Q & A

What is the origin of the word 'math' according to the script?

The word 'math' is derived from a Greek word that means knowledge.

What is the modern definition of math mentioned in the script?

Math is defined as the field that deals with amount, structure, space, and change.

Why does the script suggest starting with simple examples to understand math?

Starting with simple examples is a method mathematicians use to help process the general idea of math, as it provides concrete starting points for abstract concepts.

What is the first thing the script mentions that math deals with?

The first thing math deals with, as mentioned in the script, is numbers.

What are the different types of numbers discussed in the script?

The script discusses natural numbers, zero, negative numbers, rational numbers, real numbers, and complex numbers.

What is the significance of the number 0 in the script's discussion of numbers?

The script points out that while zero is taken for granted today, it was not always accepted or understood throughout most of human history.

What mathematical operations are mentioned in the script as something we learn from an early age?

The script mentions addition, subtraction, multiplication, and division as mathematical operations we learn in early childhood.

What is the script's explanation of the term 'function' in the context of math?

In the script, a function in math is likened to a process in computer science where an input is taken, processed, and an output is produced.

How does the script describe the concept of 'relations' in math?

The script describes relations as comparisons between two numbers, such as when comparing the heights of the Everest and the Dead Sea.

What is the role of 'ordered pairs' in the script's discussion of math?

Ordered pairs are used to describe points on Earth, such as longitude and latitude, and are highlighted as an object of interest in how math formally describes order and structure.

What is the most basic mathematical object according to the script's introduction of Set Theory?

According to the script, the most basic object in Set Theory is a set, which is used to construct other basic objects in math.

Outlines

00:00

📚 Introduction to Basic Math Concepts

The course will cover basic math concepts, starting from scratch. It begins with exploring the definition of math, tracing its origins from the Greek word for knowledge. Modern definitions describe math as dealing with amount, structure, space, and change, but the course will simplify this by using basic examples to illustrate these ideas. The first key topic is numbers, including natural numbers, zero, negative numbers, and rational numbers, as well as more complex numbers like π (pi) and the golden ratio, leading to real numbers. The course will not cover complex numbers. Understanding numbers involves learning operations such as addition, subtraction, multiplication, and division, and their role in functions, which will be explored later. Numbers also help in comparisons, relations, and organizing data, such as coordinates for geographical locations. The fundamental mathematical concept to be introduced is the set, which will be used to build other mathematical objects.

05:00

🔢 Building Math with Sets

The course will focus on constructing various mathematical objects starting from the concept of a set. This includes creating numbers, ordered pairs, relations, and functions. The introduction to set theory will serve as a foundation for understanding these elements. The course will explore how sets can help structure and interrelate mathematical terms and objects. The goal is to provide a basic understanding of these foundational concepts to facilitate further study in mathematics. The session concludes with encouragement and a sign-off, indicating the end of the introduction and the beginning of the journey into set theory and its applications.

Keywords

💡Math

Math, short for mathematics, is a field that deals with the properties and relationships of numbers, quantities, shapes, and patterns. In the video, math is described as a discipline that extends beyond just numbers, encompassing structure, space, and change. It is foundational to the course, which aims to explore math from a very basic level, using simple examples to convey complex ideas.

💡Natural Numbers

Natural numbers are the set of positive integers starting from 1, such as 1, 2, 3, and so on. They are the most basic and intuitive numbers we learn about in childhood. In the video, natural numbers are introduced as a starting point for understanding the broader concept of numbers in mathematics.

💡Zero

Zero is a unique number that represents the absence of quantity. It is considered a part of whole numbers but was not universally recognized in early human history. The video script notes that while we take zero for granted today, it was not always a part of human understanding of numbers.

💡Negative Numbers

Negative numbers are numbers that are less than zero, such as -1, -17, etc. They are used to represent quantities in debt or temperatures below zero. The script mentions that negative numbers, while widely accepted today, were not as readily understood or accepted throughout most of human history.

💡Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. Examples include halves (1/2) and two-thirds (2/3). The video script describes rational numbers as a category of numbers that includes fractions, which are formally recognized in mathematics.

💡Real Numbers

Real numbers include all rational numbers, irrational numbers (like pi or the golden ratio), and integers. They form a continuum that can be represented on a number line. The video script introduces real numbers as a larger system that encompasses more exotic numbers beyond natural and rational numbers.

💡Complex Numbers

Complex numbers are numbers that involve the square root of negative one, denoted by 'i'. They extend the concept of real numbers by allowing for the combination of real and imaginary parts. Although mentioned in the script, the course will not delve into complex numbers, indicating they are a more advanced topic.

💡Functions

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The video script likens functions to the process of taking an input, processing it, and producing an output, which is a fundamental concept in both mathematics and computer science.

💡Relations

Relations in mathematics describe the way in which one set of numbers or values is associated with another. The script uses the example of comparing the heights of Mount Everest and the Dead Sea to illustrate how relations can describe the comparative size or order of quantities.

💡Ordered Pairs

Ordered pairs are pairs of numbers where the order is significant, such as the coordinates for a point on Earth's surface (longitude and latitude). The video script explains that ordered pairs are important in mathematics for organizing and describing locations or positions.

💡Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. The video script suggests that set theory provides a foundational approach to building the rest of mathematics, starting with the concept of a set and using it to construct other basic mathematical objects.

Highlights

The course will cover basic math concepts and start from scratch.

Math is derived from a Greek word meaning knowledge.

Modern definitions of math include dealing with amount, structure, space, and change.

Instead of general definitions, the course will use simple examples to illustrate math concepts.

Math deals with numbers, which is a good starting point for understanding.

Natural numbers are familiar from childhood, such as 1, 3, or 42.

Zero is a number taken for granted today but wasn't always accepted in history.

Negative numbers, like -17, are now more accepted than in the past.

Fractions, referred to as rational numbers in math, include examples like half or two-thirds.

Exotic numbers like pi or the golden ratio are part of real numbers.

Complex numbers exist but won't be covered in this course.

Basic arithmetic operations include addition, subtraction, multiplication, and division.

In math, these operations are functions, which will be explored in the course.

Comparisons, such as heights of Everest and the Dead Sea, describe relations between numbers.

Pairs of numbers, like longitude and latitude, are used to describe points on Earth.

Ordered pairs help in building relations and are fundamental in math.

Functions are a special type of relation.

The course will introduce the concept of a set as the most basic mathematical object.

Set theory will be used to construct numbers, ordered pairs, relations, and functions.

Transcripts

Hello, In this course we're going to deal

with math.

And we're going to that on a very basic level of math.

I will even try to show you

how to do certain things basically from scratch.

The first thing that I want to ask,

before we dive into the technical stuff,

what is math?

If I were to open a dictionary

and try to understand the origin of the word

I will see that it is derived from a Greek word that means knowledge.

It doesn't say much about what math actually is.

If I were to read modern definitions

I will encounter definitions like:

"Math is the field that deals with amount, structure, space, and change."

It's all true, but feels very up in the air.

So instead of trying to give a general definition,

I will so what mathematicians are very fond of doing,

and start with simple examples.

Hopefully, they will help processing the general idea.

If we ask a random person on the street what is math?

There's a good chance they will say that it the field that deals with numbers.

That's absolutely right.

One of the things math deals with

is numbers, and we will deal with it in our course as well.

Math deals with many other things as well,

but numbers is a good starting point.

What are numbers?

There are things like 1, 3, or 42,

that we already understand their meaning pretty well in childhood.

Those numbers are called natural numbers.

There's 0,

which we take for granted these days,

but that wasn't the case for most of human history.

And there are numbers such as -17, called negative numbers,

which are also a lot more acceptable today

then they were for most of history.

And there are fractions, such as a half or two-thirds

In math we usually refer to these fractions

as rational numbers.

And there are more exotic numbers,

such as pie or the golden ration,

which aren't natural or even rational.

They are included in a larger system of numbers,

the real numbers.

There's also something beautiful, called complex numbers,

but we won't get to that in this course.

So we have numbers, but what do we do with the,?

From an early age we already know

to add and subtract and multiply, and maybe even divide.

What are these operations?

In computer science, it is said that we take an input

process it, and get an output.

In math, we call this thing a function.

Here, another basic mathematical term.

Functions, what are they?

We'll see that later. What else do numbers do?

Well, when it's said the the Everest is higher than the Dead Sea

we are doing a comparison.

We compare two numbers

that represent the heights of these places.

Such a comparison describes a relation between two numbers.

What are relations?

We will talk about that during the course as well.

Another thing that we do with numbers

is to organize them in different ways.

For example, every point on the face of the Earth

is described as a pair of numbers, the point's longitude and latitude.

Pairs of things, where the order matters,

is also an object that we'd want to understand

how math formally describes,

and large parts of mathematics are structured with its help.

Later on, we'll see that many of these terms

are interrelated and codependent.

For example, ordered pairs of elements

help us to build relations.

And functions are special type of a relation.

And so, the question we now ask

what is the most basic mathematical object

that we can talk about here?

An object that would help us "build" the rest of math?

The answer is that there are many different approaches

for this mathematical question, but,

there is one that currently very popular

and it's the one that I will introduce later.

According to it, the most basic object is –

a set.

Therefore, this part of the Set Theory course

will start with the concept of a set, and use it to construct

some of the other basic objects that make up math.

In particular, we will create numbers, ordered pairs, relations, and functions.

Good luck to us!

Thanks, and see you next time.