10 Amazing Math Facts You Never Learned In School

Summary

This script explores fascinating mathematical concepts like palindromic numbers, non-Euclidean geometries, the absence of zero in Roman numerals, the Möbius strip, the Four Color theorem, the Tower of Hanoi puzzle, the Birthday Paradox, the Banach-Tarski paradox, and the Mandelbrot set, showcasing the beauty and complexity of mathematics.

Takeaways

• 🔢 The number 111,111 is a palindrome, reading the same forwards and backwards, and when squared, it forms another palindromic number: 12345678987654321.
• 📐 In non-Euclidean geometries like spherical and hyperbolic geometry, the sum of angles in a triangle differs from the Euclidean 180°, being more or less depending on the geometry.
• 🌐 Spherical geometry is relevant to our understanding of the Earth, where the shortest distance between two points is along a great circle.
• 🈚️ Roman numerals lack a symbol for zero, reflecting the historical absence of zero in early mathematical systems.
• 🔍 The Möbius strip is a one-sided object with intriguing properties, challenging our understanding of surfaces in topology.
• 🎨 The Four Color Theorem, proven in 1976 with computational help, states that any map can be colored with four colors such that no two adjacent regions share the same color.
• 🧩 The Tower of Hanoi puzzle involves moving a stack of discs to another rod with specific rules and has a mathematical formula for the minimum number of moves to solve it.
• 🎲 The Birthday Paradox illustrates the surprisingly low number of people (23) needed in a room to have a greater than 50% chance of two people sharing the same birthday.
• 🪄 The Banach-Tarski Paradox, using set theory and the Axiom of Choice, shows that a solid sphere can be decomposed and reassembled into two spheres of the same size.
• 🐴 The Mandelbrot set is a famous fractal with a boundary known as the seahorse tail, characterized by a series of spirals and seahorse-shaped formations.
• 🤯 The script promises to introduce some mind-blowing numbers that are sure to amaze the audience.

Q & A

What is a palindromic number and why is 111,111 considered one?

A palindromic number is a number that remains the same when its digits are reversed. The number 111,111 is considered a palindromic number because it reads the same forwards and backwards.

What happens when you square the palindromic number 111,111?

When you square 111,111, the result is 12345678987654321, which is also a palindromic number. This demonstrates an interesting property where the square of a palindromic number can also be palindromic.

Why do the angles of a triangle in Euclidean geometry add up to 180°?

In Euclidean geometry, the angles of a triangle add up to 180° due to the flat nature of the plane on which the geometry is based. This is a fundamental property of triangles in this type of geometry.

How does spherical geometry differ from Euclidean geometry in terms of triangle angles?

In spherical geometry, the angles of a triangle add up to more than 180°. This is because spherical geometry deals with figures on the surface of a sphere, where the curvature affects the angle sum.

What is the Four Color theorem and why is it significant?

The Four Color theorem states that any map can be colored using only four colors in such a way that no two adjacent regions have the same color. It is significant because it was a long-standing problem in mathematics that was eventually proven using computer algorithms in 1976.

What is the Tower of Hanoi puzzle and how is it solved?

The Tower of Hanoi is a puzzle involving rods and discs, where the objective is to move a stack of discs from one rod to another, following the rule that only one disc can be moved at a time and no disc can be placed on top of a smaller disc. The minimum number of moves required to solve the puzzle is given by the formula 2^n - 1, where n is the number of discs.

What is the birthday paradox and what does it illustrate?

The birthday paradox illustrates the probability that at least two people in a group share the same birthday. It shows that only 23 people are needed in a room for there to be a greater than 50% chance that any two people share a birthday, which is counterintuitive.

What is the Banach-Tarski paradox and what does it involve?

The Banach-Tarski paradox is a result in set theory that shows it is possible to decompose a solid sphere into a finite number of pieces and then reassemble those pieces into two solid spheres of the same size. It is a counterintuitive result that challenges our intuitive understanding of geometry.

What is the Mandelbrot set and why is it famous?

The Mandelbrot set is a famous fractal in mathematics, characterized by its boundary, the seahorse tail, which consists of increasingly smaller spirals and seahorse-shaped formations. It is renowned for its beauty and has inspired significant research in the field.

Why is there no Roman numeral for zero?

There is no Roman numeral for zero because the Roman numeral system was an additive system that used symbols like I, V, X, L, etc. It lacked the concept of zero, which is crucial for modern mathematics and the representation of numbers.

What is a Möbius strip and what are its properties?

A Möbius strip is a geometric object with only one side and one edge, formed by giving a long rectangular strip a half-twist and attaching its ends. It has fascinating properties, such as being a continuous loop that can be traced back to its starting point without crossing an edge.

Outlines

00:00

🔢 Mathematical Curiosities and Geometry

This paragraph delves into the intriguing world of palindromic numbers and their properties when squared, showcasing the fascinating patterns they form. It also contrasts Euclidean geometry with spherical and hyperbolic geometries, highlighting the unique properties of angles in these systems. Furthermore, it touches on the historical significance of the absence of a Roman numeral for zero and introduces the Möbius strip, a one-sided geometric object with intriguing properties. The Four Color theorem and the Towers of Hanoi puzzle are also mentioned, emphasizing their impact on graph theory and mathematical problem-solving, respectively. Lastly, the paragraph presents the counterintuitive nature of the birthday paradox, challenging common assumptions about probability.

05:02

🎨 The Banach-Tarski Paradox and Mathematical Beauty

The second paragraph introduces the Banach-Tarski paradox, a mind-boggling concept from set theory that allows for a solid sphere to be decomposed and reassembled into two spheres of equal size. Despite the complexity and the humorous attempt at animating this concept, the paragraph emphasizes the beauty and allure of mathematical phenomena. It then shifts focus to the Mandelbrot set, a famous fractal known for its boundary's intricate patterns, such as the seahorse tail. The paragraph concludes by promising to reveal more astonishing numbers that will captivate the audience's imagination.

Keywords

💡Palindrome

A palindrome is a sequence of characters that reads the same forward and backward. In the script, '111,111' is mentioned as a palindrome, and its squared value '12,345,678,987,654,321' is also a palindrome. This illustrates a fascinating property in mathematics where certain numbers maintain their symmetry even through mathematical operations.

💡Spherical Geometry

Spherical geometry is the study of figures on the surface of a sphere, such as the Earth. Unlike Euclidean geometry, the angles of a triangle in spherical geometry add up to more than 180 degrees. The script uses the Earth's curvature to explain this concept, emphasizing how different types of geometry apply to different surfaces.

💡Hyperbolic Geometry

Hyperbolic geometry is a type of non-Euclidean geometry where the angles of a triangle add up to less than 180 degrees. This is compared to the shape of Pringle chips in the script. The concept illustrates how altering the surface can change fundamental geometric properties.

💡Roman Numerals

Roman numerals are an ancient numeric system using letters from the Latin alphabet. The script highlights that Roman numerals lack a symbol for zero, showcasing a historical limitation in numeric representation. This absence illustrates how mathematical concepts like zero were not always part of human knowledge.

💡Möbius Strip

A Möbius strip is a one-sided geometric object created by giving a rectangular strip of paper a half-twist and joining the ends. The script explains that drawing a line along the center of the strip eventually returns to the starting point, demonstrating its unique property of having only one side, which is a fundamental concept in topology.

💡Four Color Theorem

The Four Color Theorem states that any map can be colored using only four colors in such a way that no two adjacent regions share the same color. Proven in 1976 using computer power, the theorem had a significant impact on graph theory and combinatorics. The script discusses the initial skepticism and eventual acceptance of this theorem.

💡Towers of Hanoi

The Towers of Hanoi is a mathematical puzzle involving moving a stack of disks from one rod to another, following specific rules. The script describes the puzzle's simplicity and complexity, noting the formula for the minimum number of moves required to solve it, which is 2^n - 1 where n is the number of disks.

💡Birthday Paradox

The Birthday Paradox illustrates that in a group of 23 people, there's a greater than 50% chance that two people will share the same birthday. This counterintuitive result from probability theory is explained in the script, highlighting common misconceptions about estimating probabilities.

💡Banach-Tarski Paradox

The Banach-Tarski Paradox states that a solid sphere can be divided into a finite number of pieces that can be reassembled into two identical spheres. The script mentions this paradox to illustrate a mind-boggling result in set theory and the axiom of choice, which challenges intuitive notions of volume and geometry.

💡Mandelbrot Set

The Mandelbrot set is a famous fractal defined by a set of complex numbers that produces intricate, self-similar patterns. The script describes the 'seahorse tail' region of the Mandelbrot set, emphasizing its aesthetic and mathematical significance, as well as its impact on the study of fractals.

Highlights

111,111 is a palindrome and squaring it results in another palindrome: 12,345,678,987,654,321.

121 is a palindrome and when squared, it remains a palindrome.

In Euclidean geometry, the angles of a triangle add up to 180 degrees.

In spherical geometry, the angles of a triangle add up to more than 180 degrees.

In hyperbolic geometry, the angles of a triangle add up to less than 180 degrees.

There's no Roman numeral for zero, highlighting that zero wasn't always accepted in mathematics.

The Möbius strip is a one-sided object with unique properties and is a great introduction to topology.

The Four Color Theorem states that any map can be colored using only four colors so that no two adjacent regions have the same color.

The Towers of Hanoi puzzle can be solved in a minimum of 2^n - 1 moves, where n is the number of discs.

The Birthday Paradox states that in a room of 23 people, there is a greater than 50% chance that two people share the same birthday.

The Banach-Tarski Paradox states that a solid sphere can be decomposed into finite pieces and reassembled into two identical spheres.

The Mandelbrot set features a fascinating boundary called the Seahorse Tail, characterized by a series of increasingly smaller spirals and seahorse-shaped formations.

Spherical geometry is used to study figures on the surface of a sphere, like the Earth.

Hyperbolic surfaces, like Pringle chips, illustrate hyperbolic geometry where the angles of a triangle are less than 180 degrees.

The Roman numeral system was additive and lacked the mathematical properties we use today, such as the concept of zero.

Transcripts

111 mil

111,111 is a paland Drome it's the same

forward and backwards now that's a

little boring because it's all ones but

if we Square it it becomes a new paland

Drome that's right one two3 45 6789

87654321 is an incredible palindromic

number there are others 121 is a

palindrome and when you square it it's

also a palindrome and these are just the

first examples on my list of really cool

amazing math facts you probably never

learned in school every good geometry

student knows the angles of a triangle

add up to 180° right not always

certainly that's true if we're talking

about ukian geometry but there are other

types of geometries specifically

spherical geometry is a good

representation for our planet spherical

geometry is the study of figures on the

surface of a sphere much much like the

Earth lines are replaced with great

circles which are the largest circles

that can be drawn on the sphere's

surface the shortest distance between

two points on a sphere is along a great

circle connecting those points when

dealing with triangles in spherical

geometry we find the angles of triangles

adding up to more than

180° this is kind of understood better

if you consider the curvature of the

Earth or a sphere as weird as that might

be there's an even weirder non ukian

geometry known as hyperbolic geometry in

this geometry you guessed it the angles

of a triangle add up to less than 180°

here imagine triangles being drawn on

hyperbolic surfaces like Pringle chips

did you ever notice there's no Roman

numeral for zero that's right if you try

to write the Roman numerals you'll see

that there's no way to indicate an

absence of something this highlights an

important part of mathematics history

that zero wasn't always always accepted

the Roman numeral system was an additive

system using I's V's L's and so on it

lacked a lot of the nice mathematical

properties that we take for granted

today the mobia strip is a fascinating

geometric object take a long rectangular

piece of paper give it a Twist and

attach both ends you've just created a

one-sided object it's essentially just a

continuous loop with a Twist but it has

a ton of cool properties and it's it's a

great introduction to topology if you

were to draw a line along the center of

the strip you'd eventually return to

your starting point even though there's

in fact only one side it's a pretty

captivating geometrical object and

really kind of defies our understanding

of how surfaces and geometry works the

Four Color theorem is a famous problem

in math that states any map can be

colored using only four colors did I

mention it's in such a way that no two

adjacent regions have the same color now

it was proven in 1976 but it took a ton

of time and computational power and when

I say computational I mean computer

power and that was kind of a big deal

back in the day given that it was met

with a lot of skepticism and criticism

but now it's universally accepted and

has significantly impacted things like

graph Theory and commentator the towers

of Hanoi is a famous puzzle made up of

rods and discs the puzzle starts with

the discs in a stack in ascending order

and the objective is to move the entire

stack to another Rod only one disc can

be moved at a time each move consists of

taking the top disc from one of the

stacks and placing it on another Rod no

disc can be placed on top of a smaller

disc now the beauty behind this is in

its Simplicity yet its complexity

there's a nice mathematical formula for

the minimum number of moves to complete

the game it's 2 the N minus one where n

is the the number of discs so the

minimum number of moves to win with

three discs would be 23 minus 1 or seven

moves the birthday problem or the

birthday Paradox is a fascinating piece

of mathematical probability theory that

illustrates how bad humans are at

estimating probability and statistics

here's the question how many people need

to be in the same room such that the

probability of two people having the

same birthday is above 50% one want to

take a

guess maybe you think 100 maybe a little

bit more the answer is 23 that's right

it only takes 23 people in the same room

to have a greater than 50% chance that

any two people have the same birthday

which is extremely

counterintuitive but all you need to do

is work out the combinatorics and the

probability and there it is I think what

most people confuse this with is the

probability that you yourself has the

same birthday as someone else that's

much more unlikely but the question's

asking do you share a birthday and does

anybody else in the room share a

birthday and when you think of it like

that the odds get much higher time for

some mispronunciation the banak tarski

Paradox says that if we take a solid

sphere and decompose it into finite

pieces we can put the pieces back

together in such a way that it creates

two solid spheres of the same size how

can that be well I'm not going to get

into the details but set theory and the

Axiom of choice prove it to be so when I

was trying to make the animation for

this it did not turn out how I wanted it

to at all but I found it too hilarious

to not leave in the video so just watch

your

eyes of course the beautiful and famous

mandal BR set has captivated

mathematicians and math enthusiasts

alike forever one particularly

fascinating aspect of the mandal BR set

is the boundary called the seahorse tail

it's a region characterized by a series

of increasingly smaller Spirals and

seahorse shaped formations that extend

outward from the main body of the set

arguably the most famous fractal in the

world the mandal BR set has brought

significant Beauty and art to

mathematics and inspired a ton of

research now if you want to see

something else pretty amazing you're

going to want to check out these numbers

they will totally blow your mind I

promise