Lec 03 - Real and Complex Numbers

Summary

In this lecture, the progression from natural numbers to complex numbers is explained. The rational numbers, defined by P/Q where P and Q are integers, are dense, meaning between any two rationals there is another rational. Despite this, not all numbers are rational, such as square roots of non-perfect squares like √2, known to be irrational since ancient times. Real numbers include all rational and irrational numbers, both dense, and highlighted by constants like π and e. Complex numbers, an extension involving square roots of negatives, are noted but not covered in this course. This progression outlines the comprehensive nature of number systems.

Highlights

• The progression from natural numbers to rational numbers involves interpreting integers as ratios P/Q. 🔢
• Real numbers are an amalgamation of rational numbers and irrational numbers like √2 and π. 🌈
• Complex numbers extend real numbers to include roots of negative numbers, though they're not part of this course. 🚀

Key Takeaways

• Understanding the density of rational numbers and how they fill the number line 📏
• Introduction to irrational numbers like the square root of 2, which cannot be expressed as P/Q 🙅‍♂️➗
• Real numbers comprise both rational and irrational numbers, expanding the number line 🧮
• Famous irrational numbers include π and e, crucial in mathematics 🎲
• Complex numbers arise from the need to handle square roots of negative numbers, beyond the course's scope 🤯

Overview

In this lecture, the fascinating journey from natural numbers to rational and complex numbers was explored. We began by revisiting natural numbers and the integers, gradually introducing rational numbers that can be expressed as a ratio, P/Q. Rational numbers are densely packed on the number line, meaning between any two, another rational number exists, giving them their densely filled characteristic. But what happens when we encounter numbers that simply do not fit into the rational framework? Enter irrational numbers.

Irrational numbers, such as the square root of 2, cannot be expressed as a ratio of two integers, making them intriguing outliers on the number line. Despite this paradox, irrational numbers are very much real and measurable, as evidenced by constants such as π and e. Real numbers thus form a grand coalition of all rational and irrational entities, systematically extending the number system we know by bridging gaps left by rationality alone.

Finally, we took a brief detour into the realm of complex numbers, which are contrived to address the square roots of negative numbers, taking us beyond real numbers into a more abstract domain. While these mysterious numbers won't be part of this course, they represent an ultimate extension in the story of numbers. This lecture beautifully outlines how each category builds upon its predecessor, crafting a fascinating hierarchy of numeric understanding.

Chapters

• 00:00 - 00:30: Introduction to Numbers This chapter introduces the concept of numbers with a focus on natural numbers, integers, and rational numbers. It explains how rational numbers are defined by fractions P/Q where both P and Q are integers. The chapter also covers the property of density in rational numbers, illustrating that between any two numbers on a number line, there is always another rational number.
• 00:30 - 01:00: Rational Numbers and Their Properties The chapter 'Rational Numbers and Their Properties' explores the nature and characteristics of rational numbers. It discusses the concept of rational numbers on a number line, explaining that between any two points on a number line, there exists a rational number. Additionally, the chapter clarifies that integers are also rational numbers, as any integer can be expressed in the form of a fraction, p/q, with a denominator of 1.
• 01:00 - 01:30: Integers as Special Rational Numbers This chapter discusses how integers can be seen as special rational numbers, specifically through rational numbers that have a reduced form denominator of 1. It questions whether all rational numbers can fill a number line, leading to an exploration of the concept of square numbers and their roots.
• 01:30 - 02:00: Introduction to Square Roots The chapter introduces the concept of square roots, explaining it as the number R that, when multiplied by itself, equals another number M. It highlights finding the number that must be squared to result in M, defining this number as the square root. Examples include perfect squares like 1, 4, 9, 16, and 25, whose square roots are integers: 1 (square root of 1), 2 (square root of 4), and 5 (square root of 25).
• 02:00 - 02:30: Perfect Squares and their Square Roots This chapter titled 'Perfect Squares and their Square Roots' deals with understanding perfect squares, their square roots, and what happens when dealing with non-perfect squares. It explains that some integers are perfect squares, such as 25 and 256, with clear square roots like 5 and 16 respectively. The chapter also explores the scenario where numbers are not perfect squares, using 10 as an example. It describes how the square root of 10 is not an integer and lies between 3 and 4, raising the question of whether such square roots, for non-perfect squares, are rational or not. The discussion leads into the broader topic of understanding the nature of square roots of non-square numbers.
• 02:30 - 03:00: Non-Perfect Squares and Irrational Numbers The chapter discusses integers that are not perfect squares, starting with the smallest such number, which is 2. It is highlighted that the square root of 2 cannot be expressed as a fraction (P/Q), a fact known since ancient Greek times, particularly by Pythagoras. The explanation includes a method to illustrate this concept, possibly through geometric representation.
• 03:00 - 03:30: The Example of Square Root of Two The chapter discusses the concept of square root of two, explaining that it is a real number that can be physically represented. By applying Pythagoras's theorem, it demonstrates that the hypotenuse of a right triangle, with each side measuring 1, equals the square root of 2. This illustrates that square root of two is a tangible and real quantity, contrary to some perceptions of it as an abstract or 'unreal' number.
• 03:30 - 04:00: Visualizing Square Root of Two The chapter titled 'Visualizing Square Root of Two' explains that the square root of 2 cannot be expressed as a rational number, written as P over Q. Despite being a measurable quantity that can be visually represented as a length, the square root of 2 does not fit into the number line of rational numbers, which are dense and seem to comprise all numbers. The existence of such a number that is not rational is significant, highlighting its unique place in mathematics.
• 04:00 - 04:30: Definition of Irrational Numbers This chapter explains irrational numbers, which are numbers that extend rational numbers on the number line to form real numbers. The real numbers include all rational and irrational numbers and are represented by the symbol R. This is an expansion of the number classifications including natural numbers (N), integers (Z), and rational numbers (Q). Irrational numbers are essential in forming the real numbers (R), providing a complete set that exists on the number line.
• 04:30 - 05:00: Real Numbers The chapter discusses the property of real numbers being dense, similar to rational numbers. It explains that between any two real numbers, there exists another real number, illustrating this by taking the average of two given real numbers.
• 05:00 - 05:30: Density of Real Numbers The chapter titled 'Density of Real Numbers' introduces the concept of irrational numbers, specifically focusing on the well-known number pi. Pi is defined as the ratio of the circumference to the diameter of a circle, a constant value that remains invariant for any circle. The chapter emphasizes pi's status as an irrational number, a type of real number that cannot be expressed as a simple fraction, highlighting the density of irrational numbers within the set of real numbers.
• 05:30 - 06:00: Famous Irrational Numbers: Pi and E This chapter discusses famous irrational numbers, focusing on Pi and E. It explains that irrational numbers cannot be expressed as a fraction (P/Q) and have infinite decimal expansions. The chapter specifically highlights that Pi and the natural logarithm base E (approximately 2.718) are well-known examples of such numbers. Additionally, it mentions that the square root of non-perfect squares, like square root of 2, 3, and 6, are also irrational numbers.
• 06:00 - 06:30: Conclusion on Real Numbers In the conclusion on real numbers, it is highlighted that while some irrational numbers might not seem useful, constants like π (pi) and e are very significant. The discussion emphasizes that real numbers extend beyond rational numbers, considering the addition of irrational numbers to the number line. The narrative explores the square root operation, which introduces irrational numbers, and poses questions about the implications of this operation. This is a point of reflection on whether the exploration of numbers stops at real numbers or continues further.
• 06:30 - 07:00: Complex Numbers and Negative Square Roots This chapter discusses complex numbers and the concept of negative square roots. It revisits the sign rule for multiplication, explaining that multiplying two numbers with the same sign results in a positive product, while differing signs produce a negative result. To explore the notion of multiplying two numbers to get negative one, it highlights that one of the numbers must be negative. The content suggests a foundational step into understanding complex numbers, where the square root of negative numbers is addressed.
• 07:00 - 09:00: Summary and Conclusion The chapter discusses square roots, highlighting that square roots must yield positive numbers when multiplied by themselves. Since negative numbers cannot have real square roots, the concept of complex numbers is introduced. Complex numbers expand upon real numbers, just as real numbers build upon rational numbers, which in turn extend integers.

Lec 03 - Real and Complex Numbers Transcription

• 00:00 - 00:30 [Music] so we started with the natural numbers and the integers and then we moved on to the rational numbers which are defined as P by Q where P and Q are both integers so we decided that the rational numbers are dense right and that means that on this number line between any two
• 00:30 - 01:00 rationals you can find a rational so if I want to now talk about this number line then I know that if I take any two positions then I'll find the ration between them and everyone rational between there and so on so it makes sense to ask this question which is that if I take any two points in the rational between them any two points then is this entire number line composed only of rational numbers of course some of those rational numbers are integers so an integer is a rational number because I can write 7 for instance as 7 by 1 right so this is of the form P by Q so any
• 01:00 - 01:30 rational number which in reduced form as denominator 1 is an integer so an integer is a special case of a rational number so do all the rational numbers fill up this number line that is the question so it turns out this is not the case so remember that a square of a number is the number multiplied by self so if I take a number m and multiply it by itself I get M Squared which is M times M and if I take this operation and turn it around then the square root of a
• 01:30 - 02:00 number is that number R such that R times R is equal to M right so I want to find out which number I have to square in order to get M and that's called the square root so if we take the so-called perfect squares like 1 4 9 16 25 and so on their square roots are integers so 1 squared is 1 so the square root of 1 is 1 2 squared is 4 so the square root of 4 is 2 5 squared is 25 so square root of 5
• 02:00 - 02:30 is 25 is 5 16 squared is 256 so square root of 256 is 16 and so on so some integers are clearly squares of other integers and so you can get the square root and find any teacher now what happens if something is not a square right so supposing I take a number which is not a square like 10 and I take its square root I know that the square root is not an integer it's somewhere between 3 & 4 because 3 squared is 9 and 4 squared is 16 question is is it a rational number or not so what happens to the square roots
• 02:30 - 03:00 of integers that are not perfect squares so the smallest such number which is not a perfect square because one remember is a perfect square one times one is one the smallest such number that is not a perfect square is actually two and it is one of the very old results that the square root of two cannot be written as P by Q this was certainly known to the ancient Greeks in fact to Pythagoras and one way to do this is to see that you can actually draw a line of so this is
• 03:00 - 03:30 not an unreal number in that sense right so you can actually draw a line of this length because if you take a square whose sides are 1 right so this is 1 then if you remember your Pythagoras's theorem then the hypotenuse of this triangle is going to be square root of 1 + 1 1 squared plus 1 squared technically which is square root of 2 so I can actually physically draw a line whose length is square root of 2 so this is a very real quantity on the other hand for
• 03:30 - 04:00 reasons that we will not describe here but there will be a separate lecture explaining this for if you are interested square root of 2 cannot be written as a rational number P by Q so here is a number which is a very measurable quantity I can actually draw this quantity as a length at the same time it does not fit into this number line of rational numbers which seems to cover all the rational numbers all the numbers because they are dense so square root of 2 since it is not a rational number and yet it exists is called an
• 04:00 - 04:30 irrational number and these numbers which constitute all the rational numbers and the real irrational numbers together are called the real numbers so the real numbers are denoted by this double line R so we had n for the natural numbers Z for the integers Q for the rational numbers and now we have the real numbers R so the real numbers extend the rational numbers by these so-called irrational numbers which are very much on the number line but which
• 04:30 - 05:00 cannot be written in the form P by Q now it's not difficult to argue that like the rationals the real numbers are dense for the very same reason because if you have two real numbers R and R prime such that R is smaller than R Prime then you can just take their average R plus R prime divided by two this must be a number which is bigger than R and it is smaller than R Prime and therefore it must lie between them so between any two real numbers you will find another real number so the real numbers are also
• 05:00 - 05:30 dense so there are some irrational numbers which we use a lot in mathematics and which you have probably come across one of them is this famous number pi which comes when we are talking about circles because it is the ratio of the circumference to the diameter okay and this is an invariant pi is always the diameter the circumference divided by the diameter for any circle is pi okay so pi is an irrational number we cannot
• 05:30 - 06:00 write it in the form P by Q and it has this if we write it in this decimal form it has this infinite decimal expansion another number which is very popular as an irrational number is this number E which is used for natural logarithms so it is two point seven one eight two eight one eight and so on right so there are a lot of rational numbers so square root of two as we have seen as a rational number it will turn out that square root of anything square root of three is also an irrational number square root of six is also an irrational number anything which is not a perfect
• 06:00 - 06:30 square its square root is actually in irrational number but many of these numbers are not very useful to us but PI and E are certainly very useful irrational numbers so now we have seen that we can find more numbers on the night line than just the rationals and these are the real numbers so do we stop here well let's look at the square root operation which we use in order to claim that there are irrational numbers so what happens if we now take the square
• 06:30 - 07:00 root of a negative number like minus 1 so remember that we had a sign rule for multiplication the sign rule for multiplication said that if I multiply any two numbers then if the two signs are the same that is there two negative signs or two positive signs I will get a positive sign and the answer only if the two signs are different if I have one minus sign in one plus sign will I get a negative answer so if I want to multiply two numbers and get a minus one one of them must be negative and one must not be negative but by definition a
• 07:00 - 07:30 square root is a number which is multiplied by itself the same number has to be multiplied by itself so it will have the same sign so any square root which multiplies by itself must give me a positive number so if I take a negative number there is no way to find a square root for it so if we want to find square roots for negative numbers we have to create yet another class of numbers called complex numbers so complex numbers extend the real numbers just like real numbers extend the rational numbers and rational numbers extend the integers and so on but the
• 07:30 - 08:00 good news for you is that we don't have to look at complex numbers for this course so to summarize real numbers extend the rational numbers by adding the so-called irrational numbers which cannot be represented of the form P by Q and a typical example of an irrational number is the square root of an integer that is not a perfect square so square root of 2 for example is not a rational number and this is also the case was square root of 3 square root of 5 square root of 6 and so on so except for the perfect squares none of the square roots
• 08:00 - 08:30 are actually rational numbers now just like we said that the rational numbers are dense because the average of any two rational numbers is a rational number similarly the real numbers are dense because the average of any two real numbers is a real number so we have a progression in terms of numbers so every natural number that we started with is also an integer because the integers extend the natural numbers with negative quantities now every integer is also a rational number because we can think of every integer as a ratio P by Q where
• 08:30 - 09:00 the denominator is 1 and finally every rational number is a real number because we said that the real numbers include all the rational numbers plus all the irrational numbers and finally we said that there are even things beyond rational numbers like complex numbers but we won't discuss them