Joy of X
Chapter 1
From Fish to Infinity
Numbers as shortcuts.
But they have a cost of abstraction. (They seem to exist in a level above our reality)
There is a creative process in adding and numbering.
“Math always involves both invention and discovery: we invent concepts but discover their consequences."
Numbers as shortcuts.
But they have a cost of abstraction. (They seem to exist in a level above our reality)
There is a creative process in adding and numbering.
- Numbers: shortcuts for counting by ones.
- Addition: shortcut for counting any amount.
“Math always involves both invention and discovery: we invent concepts but discover their consequences."
Chapter 2
Rock Groups
Sums of odd numbers = squares (consecutives)
Sometimes to solve things we can look at creativity; by trying to look at things differently, by a playful side.
Sums of odd numbers = squares (consecutives)
Sometimes to solve things we can look at creativity; by trying to look at things differently, by a playful side.
Chapter 3
The Enimy of My Enimy
Negative numbers exist only in our minds.
If you have 2 cookies, and I want 6 cookies, you can’t really have negative 4 cookies. We can only think of negative 4 cookies.
“Subtraction forces us to expand our conception of what numbers are. Negative numbers are a lot more abstract than positive numbers.”
Many people try to avoid negative numbers for this reason.
Why does (-1) * (-1) = +1?
Because the enemy of my enemy is my friend:
(-1) * (-1) * (-1) * (-1) = +1
Negative numbers exist only in our minds.
If you have 2 cookies, and I want 6 cookies, you can’t really have negative 4 cookies. We can only think of negative 4 cookies.
“Subtraction forces us to expand our conception of what numbers are. Negative numbers are a lot more abstract than positive numbers.”
Many people try to avoid negative numbers for this reason.
Why does (-1) * (-1) = +1?
Because the enemy of my enemy is my friend:
(-1) * (-1) * (-1) * (-1) = +1
Chapter 4
Commuting
“Every decade or so a new approach to teaching math comes along and creates fresh opportunities for parents to feel inadequate.”
Example: parents who can’t help their kids’ second grade homework.
There is a certain ambiguity in the language or in multiplications and divisions.
Commutative law:
Things equal the same thing if they are turned around.
(7*3) = (3*7)
“Every decade or so a new approach to teaching math comes along and creates fresh opportunities for parents to feel inadequate.”
Example: parents who can’t help their kids’ second grade homework.
There is a certain ambiguity in the language or in multiplications and divisions.
Commutative law:
Things equal the same thing if they are turned around.
(7*3) = (3*7)
Chapter 5
Division and It's Contents
Integers: numbers (rational numbers) + 0 + negative numbers.
Fractions: rations of integers.
“Fractions always yield decimals that terminate or eventually repeat periodically – that can be proven – and since this decimal does neither, it can’t be equal to the ratio of any whole number. It’s irrational.”
Integers: numbers (rational numbers) + 0 + negative numbers.
Fractions: rations of integers.
“Fractions always yield decimals that terminate or eventually repeat periodically – that can be proven – and since this decimal does neither, it can’t be equal to the ratio of any whole number. It’s irrational.”
Chapter 6
Location, Location, Location
Ways to represent numbers:
“What nearly all of these systems have in common is that our biology is deeply embedded in them.”
Hindu-Arabic: based on 10. 10 isn’t a single symbol reserved for 10, it’s a position.
Place-value system: “All numbers can be expressed with the same ten digits, merely by slotting them into the right places.”
This made arithmetic available to everyone. All one has to do is learn the multiplication table and its counterpart for addition.
It is a mechanical process.
But without 0 this system would collapse.
Ways to represent numbers:
“What nearly all of these systems have in common is that our biology is deeply embedded in them.”
- Tallies
- Roman numerals
- Babylonian
- Hindu-Arabic
Hindu-Arabic: based on 10. 10 isn’t a single symbol reserved for 10, it’s a position.
Place-value system: “All numbers can be expressed with the same ten digits, merely by slotting them into the right places.”
This made arithmetic available to everyone. All one has to do is learn the multiplication table and its counterpart for addition.
It is a mechanical process.
But without 0 this system would collapse.
Chapter 7
The Joy of X
Algebra: solving for x and working with formulas.
Solving for x is like a detective work. You get a few clues and you have to identify x from the information given.
“Working with formulas, by contrast, is a blend or art and science. Instead of dwelling on a particular x, you’re manipulating and massaging relationships that continue to hold even as the numbers in them change.”
Algebra: solving for x and working with formulas.
Solving for x is like a detective work. You get a few clues and you have to identify x from the information given.
“Working with formulas, by contrast, is a blend or art and science. Instead of dwelling on a particular x, you’re manipulating and massaging relationships that continue to hold even as the numbers in them change.”
Chapter 8
Finding Your Roots
“For more than 2,500 years, mathematicians have been obsessed with solving for x. the story of their struggle is to find roots, - the solutions – of increasingly complicated equations is one of the great epics in the history of human thought.”
Some of the mental blockages were:
From this (- ) came imaginary numbers.
i2 = -1
i can’t be found in the number line. It is at right angles with the number line and when you fuse the imaginary axis with the number line, you create a 2D space.
These two together are called complex numbers.
“For more than 2,500 years, mathematicians have been obsessed with solving for x. the story of their struggle is to find roots, - the solutions – of increasingly complicated equations is one of the great epics in the history of human thought.”
Some of the mental blockages were:
- Cube root of 2
- Square roots of the negatives
From this (- ) came imaginary numbers.
i2 = -1
i can’t be found in the number line. It is at right angles with the number line and when you fuse the imaginary axis with the number line, you create a 2D space.
These two together are called complex numbers.
Chapter 9
My Tub Runneth Over
Sometimes we see faulty patterns in things and come up with wrong answers.
But it is good to learn from these answers.
“Perhaps even more important, word problems give us practice in thinking not just about numbers, but about relationships between numbers.”
“Word problems initiate us unto this way of thinking.”
Sometimes we see faulty patterns in things and come up with wrong answers.
But it is good to learn from these answers.
“Perhaps even more important, word problems give us practice in thinking not just about numbers, but about relationships between numbers.”
“Word problems initiate us unto this way of thinking.”
Chapter 10
Working Your Quads
Quadratic formula:
Quadratic formula:
“It’s only when you understand what the quadratic formula is trying to do that you can begin to appreciate its inner beauty.”
When we know the relationship between the known and the unknown we are able to find the answer.
“What’s so remarkable about this formula is how brutally explicit and comprehensive it is. There’s the answer, right there, no matter what a, b, and c happen to be. Considering that there are infinitely many possible choices for each of them, that’s a lot for a single formula to manage.”
When we know the relationship between the known and the unknown we are able to find the answer.
“What’s so remarkable about this formula is how brutally explicit and comprehensive it is. There’s the answer, right there, no matter what a, b, and c happen to be. Considering that there are infinitely many possible choices for each of them, that’s a lot for a single formula to manage.”
Chapter 11
Power Tools
We unconsciously use most of the formulas and functions that we see in school.
Functions are tools that transform things:
Exponential functions (xh): parabolas and constants
Power function: building blocks of scientists and engineers to describe the growth and decay. (10x)
“As the numbers inside the logarithms grew multiplicatively, increasing tenfold, each time from 100 to 1,000 to 10,000 their logarithms grew additively, increasing from 2 to 3 to 4.”
We unconsciously use most of the formulas and functions that we see in school.
Functions are tools that transform things:
- Tools
- Building blocks
Exponential functions (xh): parabolas and constants
Power function: building blocks of scientists and engineers to describe the growth and decay. (10x)
“As the numbers inside the logarithms grew multiplicatively, increasing tenfold, each time from 100 to 1,000 to 10,000 their logarithms grew additively, increasing from 2 to 3 to 4.”
Chapter 12
Square Dancing
Geometry marvels logic and intuition
Pythagorean Theorem:
Right triangle (what you get when you cut a rectangle in half along its diagonal.
“The Pythagorean theorem tells you how long the diagonal is compared to the sides of the rectangle.”
The Pythagorean Theorem reveals the truth about the nature of space.
Geometry marvels logic and intuition
Pythagorean Theorem:
Right triangle (what you get when you cut a rectangle in half along its diagonal.
“The Pythagorean theorem tells you how long the diagonal is compared to the sides of the rectangle.”
The Pythagorean Theorem reveals the truth about the nature of space.
Chapter 13
Something From Nothing
“Every math course contains at least one notoriously difficult topic. In arithmetic, its long division. In algebra, it word problems. And in geometry, its proofs.”
Geometry is good for the mind; it trains you to think clearly and logically.
“What is important is the axiomatic method, the process of building a rigorous argument, step by step, until a desired conclusion has been established.”
“Every math course contains at least one notoriously difficult topic. In arithmetic, its long division. In algebra, it word problems. And in geometry, its proofs.”
Geometry is good for the mind; it trains you to think clearly and logically.
“What is important is the axiomatic method, the process of building a rigorous argument, step by step, until a desired conclusion has been established.”
Chapter 14
The Conic Conspiracy
Ellipses have two particular points inside: foci.
“Mathematicians and conspiracy theorists have this much in common: we’re suspicious of coincidences, especially convenient ones.”
Parabola: the set of all point equidistant from a given point and a given line not containing that point.
Ellipse: a set of point the sum of whose distances from two given points is a constant.
Both of these are cross sections of the surface of a cone.
(What is the universe is a conic section?)
Ellipses have two particular points inside: foci.
“Mathematicians and conspiracy theorists have this much in common: we’re suspicious of coincidences, especially convenient ones.”
Parabola: the set of all point equidistant from a given point and a given line not containing that point.
Ellipse: a set of point the sum of whose distances from two given points is a constant.
Both of these are cross sections of the surface of a cone.
(What is the universe is a conic section?)
Chapter 15
SIn Qua Non
Trigonometry is the key to the mathematics of circles (or sine curve)
a repeats itself each time a changes 360
“The ripples on a pond, the ridges of sand dunes, the stripes of a zebra – all are manifestations of nature’s most basic mechanism of pattern formation: the emergence of sinusoidal structure from a background of bland uniformity.”
Trigonometry is the key to the mathematics of circles (or sine curve)
a repeats itself each time a changes 360
“The ripples on a pond, the ridges of sand dunes, the stripes of a zebra – all are manifestations of nature’s most basic mechanism of pattern formation: the emergence of sinusoidal structure from a background of bland uniformity.”
Chapter 16
Take It To The Limit
Archimedes came close to inventing calculus, almost 2,000 years before Newton and Leibniz.
Pi: ratio of two distances; the diameter and the circumference.
Pi: circumference / diameter
“The key is to thinking mechanically about curved shapes, to pretend they are made up of lots of little straight pieces.” As long as we imagine infinitely many straight pieces, infinitesimally small.
Archimedes came close to inventing calculus, almost 2,000 years before Newton and Leibniz.
Pi: ratio of two distances; the diameter and the circumference.
Pi: circumference / diameter
“The key is to thinking mechanically about curved shapes, to pretend they are made up of lots of little straight pieces.” As long as we imagine infinitely many straight pieces, infinitesimally small.
Chapter 17
Change We Can Believe In
“Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curve ball.”
Calculus divides into:
Derivatives can be positive, negative or zero (rising, falling or leveling off).
“Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curve ball.”
Calculus divides into:
- Differential: derivative; how fast something is changing (rate of change)
- Integral: how much it is accumulating
Derivatives can be positive, negative or zero (rising, falling or leveling off).
Chapter 18
It Slices, It Dices
The integral sign means summation, made by Leibniz.
Integral calculus functions when we want to know the commutative changes that are non-uniform.
“If you integrate the derivative of a function from one point to another, you get the change in function between two points.”
The integral sign means summation, made by Leibniz.
Integral calculus functions when we want to know the commutative changes that are non-uniform.
“If you integrate the derivative of a function from one point to another, you get the change in function between two points.”
Chapter 19
All About e
e = 2.71828
it is the limiting number approaches by the sum as we take more and more terms.
e = 2.71828
it is the limiting number approaches by the sum as we take more and more terms.
Chapter 20
Loves Me, Loves Me Not
Differential equations:
“They represent the most powerful tool humanity has ever created for making sense of the material world.”
Physics is always expressed in differential equations.
Differential equations:
“They represent the most powerful tool humanity has ever created for making sense of the material world.”
Physics is always expressed in differential equations.
Chapter 21
Step Into the Light
Vector calculus: describes the invisible fields all around us. Describes the vectors that change.
Vector: step that carries you from one place to another.
Vectors show direction and magnitude.
Vector calculus: describes the invisible fields all around us. Describes the vectors that change.
Vector: step that carries you from one place to another.
Vectors show direction and magnitude.
Chapter 22
The New Normal
Statistics
One of the central lessons: distribution.
Things that seem hopelessly random and unpredictable when viewed in isolation often turn out to be lawful and predictable when viewed in aggregate.
Statistics
One of the central lessons: distribution.
Things that seem hopelessly random and unpredictable when viewed in isolation often turn out to be lawful and predictable when viewed in aggregate.
Chapter 23
Chances Are
Conditional probability: the probability that some event A happens, given the occurrence of some other event B.
Conditional probability: the probability that some event A happens, given the occurrence of some other event B.
Chapter 24
Untangling the Web
Linear algebra vs. search systems.
Linear algebra is called lines because it is related to lines.
Linear algebra vs. search systems.
Linear algebra is called lines because it is related to lines.
Chapter 25
The Loneliest Numbers
Prime numbers: divisible only by one and themselves.
Twin primes: prime numbers that are close to each other but always have an even number between them. 11 and 13. 17 and 19. 41 and 43.
But the loneliest number is 1.
One is excluded from primes for the sake of the theorem. The theorem: any number can be factored into primes in a unique way.
After 1 it is 2.
It is the only even prime.
Prime numbers: divisible only by one and themselves.
Twin primes: prime numbers that are close to each other but always have an even number between them. 11 and 13. 17 and 19. 41 and 43.
But the loneliest number is 1.
One is excluded from primes for the sake of the theorem. The theorem: any number can be factored into primes in a unique way.
After 1 it is 2.
It is the only even prime.
Chapter 26
Group Think
Group theory:
Symmetries of shape, leaving them unchanged.
Looking at all the possible ways for doing something.
Group theory:
Symmetries of shape, leaving them unchanged.
- Abstract
- Focus on what you can do to a shape without changing everything
Looking at all the possible ways for doing something.
Chapter 27
Twist and Shout
Topology: (an offshoot of geometry). Two shapes are regarded as the same if you can bend, twist, stretch or deform one into the other continuously without ripping of puncturing.
Objects behave as if they were infinitely elastic. (Mobius strip)
Topology: (an offshoot of geometry). Two shapes are regarded as the same if you can bend, twist, stretch or deform one into the other continuously without ripping of puncturing.
Objects behave as if they were infinitely elastic. (Mobius strip)
Chapter 28
Think Globally
Arcs of great circles: contain the shortest paths between any two points of a sphere.
Called great because they are the largest circles you can have in a sphere.
Differential geometry: studies the effects of small local differences on various kinds of shapes.
Arcs of great circles: contain the shortest paths between any two points of a sphere.
Called great because they are the largest circles you can have in a sphere.
Differential geometry: studies the effects of small local differences on various kinds of shapes.
Chapter 29
Analyze This!
Partial sum: running total.
Convergence: partial sums don’t approach any limiting value as more and more terms are included in the sum.
Partial sum: running total.
Convergence: partial sums don’t approach any limiting value as more and more terms are included in the sum.
Chapter 30
Infinite is neither off nor even.
There are as manyt positive fractionas as natural numbers.
There are as manyt positive fractionas as natural numbers.