Chapter 1- Numbers
They apparently exist in some sort of Platonic real, a level above reality.
Just as numbers are a shortcut for counting by ones, addition is a shortcut for counting by any amount. This is how mathematics grows. The right abstraction leads to new insight, and new power.
Logic leaves us no choice. Math always involves both invention and discovery: we invent the concepts but discover their consequences. In mathematics our freedom lies in the questions we ask- and in how we pursue them- but not in the answers awaiting us.
Just as numbers are a shortcut for counting by ones, addition is a shortcut for counting by any amount. This is how mathematics grows. The right abstraction leads to new insight, and new power.
Logic leaves us no choice. Math always involves both invention and discovery: we invent the concepts but discover their consequences. In mathematics our freedom lies in the questions we ask- and in how we pursue them- but not in the answers awaiting us.
Chapter 2 Rock Groups
Yet some numbers, like 2,3,5 and 7, truly are hopeless. None of them can torm any sort of rectangle at all, other than a simple line of rocks (a single row). These strangely inflexible numbers are the famous prime numbers.
But what possible connection could there be between odd numbers, with their ungainly appendages, and the classically symmetrical numbers that form squares?
The key is to recognize that odd numbers can make L-Shapes, with their protuberances cast off into the corner. And when you stack successive L-shapes together, you get a square!
But what possible connection could there be between odd numbers, with their ungainly appendages, and the classically symmetrical numbers that form squares?
The key is to recognize that odd numbers can make L-Shapes, with their protuberances cast off into the corner. And when you stack successive L-shapes together, you get a square!
Chapter 3 The Enemy of my Enemy
Subtraction forces us to expand our conception of what numbers are. Negative numbers are a lot more abstract than positive numbers.
The point is that some part of what we observe is due to nothing more than the primitive logic of the enemy of my enemy and this part is captured perfectly by the multiplication of negative numbers. By sorting the meaningful from the generic, the arithmetic of negative numbers can help us see where the real puzzles lie.
The point is that some part of what we observe is due to nothing more than the primitive logic of the enemy of my enemy and this part is captured perfectly by the multiplication of negative numbers. By sorting the meaningful from the generic, the arithmetic of negative numbers can help us see where the real puzzles lie.
Chapter 4 Commuting
If you regard multiplication as being synonymous with repeated counting by a certain number, the commutative law isn’t transparent.
It becomes more intuitive if you conceive of multiplication visually.
It becomes more intuitive if you conceive of multiplication visually.
Chapter 5 Division and Discontents
These are ratios of intgers-hence their technical name, rational numbers.
There is no way to express this decimal as a fraction. 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 numbers. It’s irrational
There is no way to express this decimal as a fraction. 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 numbers. It’s irrational
Chapter 6 Location, Location, Location
Because of its promiscuous divisibility, 60 is much more congenial than 10 for any sort of calculation or measurement that involves cutting things into equal parts. When we divide an hour into 60 minutes, or a minute into 60 seconds, or a full circle into 360 degrees, we’re channeling the sages of ancient Babylon.
All place-value systems are based on some number called, appropriately enough, the base. Our system is base 10, or decimal.
All place-value systems are based on some number called, appropriately enough, the base. Our system is base 10, or decimal.
Chapter 7 The Joy of X
Algebra- solving for x and working with formulas
The goal is to identify x from the information given
Changing numbers are called variables, and they are what truly distinguishes algebra from arithmetic. The formulas in question might express elegant patterns about number for their own sake. This is where algebra meets art. Or they might express relationships between numbers in the real world, as they do in the laws of nature for falling objects or planetary orbits or genetic frequencies in a population. This is where algebra meets science.
Another kind of formula is known as identity.
The goal is to identify x from the information given
Changing numbers are called variables, and they are what truly distinguishes algebra from arithmetic. The formulas in question might express elegant patterns about number for their own sake. This is where algebra meets art. Or they might express relationships between numbers in the real world, as they do in the laws of nature for falling objects or planetary orbits or genetic frequencies in a population. This is where algebra meets science.
Another kind of formula is known as identity.
Chapter 8 Find your ROOTS
The freewheeling use of square roots eventually forced the universe of numbers to expand..again. -1 still I for imaginary
Complex means two types of numbers, real and imaginary, have bonded together to form a complex, a hybrid number like 2+3i
Fundamental theorem of algebra says that the roots of any polynomial are always complex numbers.
The method takes a starting point and does a certain computation that improves it. By doing this repeatedly, always using the previous point to generate a better one, the method bootstraps its way forward and rapidly homes in on a root.
Wherever two colors met, the third would always insert itself and join them
Magnifying the boundaries revealed patterns within patterns.
Imperceptible changes in the initial conditions-could make all the difference
Hubbard’s work was an early foray into what’s now called Complex dynamics, chaos theory, analysis and fractal geometry.
Complex means two types of numbers, real and imaginary, have bonded together to form a complex, a hybrid number like 2+3i
Fundamental theorem of algebra says that the roots of any polynomial are always complex numbers.
The method takes a starting point and does a certain computation that improves it. By doing this repeatedly, always using the previous point to generate a better one, the method bootstraps its way forward and rapidly homes in on a root.
Wherever two colors met, the third would always insert itself and join them
Magnifying the boundaries revealed patterns within patterns.
Imperceptible changes in the initial conditions-could make all the difference
Hubbard’s work was an early foray into what’s now called Complex dynamics, chaos theory, analysis and fractal geometry.
Chapter 9 My Tub Runneth Over
There are broader lessons to be learned here-lessons about how to solve problems approximately when you can’t solve them exactly, and how to solve them intuitively, for the pleasure of the Aha! Moment.
The silver lining is that even wrong answers can be educational.. as long as you realize they’re wrong.
The undistracted reasoning that this problem requires (painitng) is one of the most valuable things about world problems. They force us to pause and think, often in unfamiliar ways. They give us practice in being mindful.
The silver lining is that even wrong answers can be educational.. as long as you realize they’re wrong.
The undistracted reasoning that this problem requires (painitng) is one of the most valuable things about world problems. They force us to pause and think, often in unfamiliar ways. They give us practice in being mindful.
Chapter 10 Working your Quads
Quadratic formula: x equals negative b, plus or minus the square root of b squared minus four a c, all over two a.
Such problems were posed in words, not symbols.
The idea is to interpret each of the terms in the equation geometrically.
Such problems were posed in words, not symbols.
The idea is to interpret each of the terms in the equation geometrically.
Chapter 11 Power Tools
Logarithms:
The bowed shape of the curve is due to the action of mathematical pliers
The arching curve above- technically known as a parabola- is a signature of the squaring function x2 operating behind the scenes.
For a parabola, n=2, constant n=0
Linear function because its xy graph is a line
Inverse square: x2 appears in the denominator.
10^x
Power of x is a variable
Base 10 is a constant
The bowed shape of the curve is due to the action of mathematical pliers
The arching curve above- technically known as a parabola- is a signature of the squaring function x2 operating behind the scenes.
For a parabola, n=2, constant n=0
Linear function because its xy graph is a line
Inverse square: x2 appears in the denominator.
10^x
Power of x is a variable
Base 10 is a constant
Chapter 12 Square Dancing
Geometry marries logic and intuition
A2+b2=c2 Pythagorean theorem
Concerned with right angle (90 degree) triangles.
Diagonal called hypotenuse.
NATURALEZA DEL ESPACIO
A2+b2=c2 Pythagorean theorem
Concerned with right angle (90 degree) triangles.
Diagonal called hypotenuse.
NATURALEZA DEL ESPACIO
Chapter 13 Something from Nothing
Arithmetic: long division
Algebra: word problems
Geometry: proofs
Axiomatic method: the process of building a rigorous argument, step by step, until a desired conclusion has been established.
Geometry as the class where they learned to be logical and creative
Algebra: word problems
Geometry: proofs
Axiomatic method: the process of building a rigorous argument, step by step, until a desired conclusion has been established.
Geometry as the class where they learned to be logical and creative
Chapter 14 The Conic Conspiracy
Parabolic curves and surfaces have an impressive focusing power of their own: each can take parallel incoming waves and focus them at a single point.
Parabola as the set of all points equidistant from a given point and a given line not containing that point.
Ellipse is the set of points the sum of whose distances from two given point is a constant.
Parabola as the set of all points equidistant from a given point and a given line not containing that point.
Ellipse is the set of points the sum of whose distances from two given point is a constant.
Chapter 15 Sine Qua Non
Without sine waves which are the atoms of structure, or nature’s building blocks there wouldn’t be nothing, giving new meaning to the phrase sine qua non
Chapter 16 Take it To the Limit
The thought of getting closer and closer and yet never quite making it.
Pi is a ratio of two distances. One of them is the diameter, the distance across the circle through its center. The other is the circumference, the distance around the circle.
Pi is defined as the ratio, the circumference divided by the diameter
A= pi r 2
Pi is a ratio of two distances. One of them is the diameter, the distance across the circle through its center. The other is the circumference, the distance around the circle.
Pi is defined as the ratio, the circumference divided by the diameter
A= pi r 2
Chapter 17 Change we can Believe in
Calculus is the mathematics of change
Two ideas are the differential and integral calculus
The derivative (differential) tells you how fast something is changing. The integral tells you how much it’s accumulating.
La logica es algo que surge con la evolucion del ser humano, evoluciona una logica, cerebro evoluciona para descubrir una logica que esta en el mundo?
The zero derivative property of peaks and troughs underlies some o the most practical application of calculus. It allows us to use derivatives to figure out where a function reaches its maximum or minumum, best or cheapest way to do something.
Differential calculus it’s a certain specific compromise between the two paths considered.
Pythagorean theorem, as well as distance equals rate times time.
Snell’s law: describes how light rays bend when they pass from air into water, as they do when the sun shines into a swimming pool
Light moves more slowly in water, and it bends accordingly to minimize its travel time.
The eerie point is that light behaves as if it were considering all possible paths and then taking the best one. Nature somehow knows calculus.
Cuando un limite tiende a dos cosas, no existe. Tiene que tender a una cosa.
Two ideas are the differential and integral calculus
The derivative (differential) tells you how fast something is changing. The integral tells you how much it’s accumulating.
La logica es algo que surge con la evolucion del ser humano, evoluciona una logica, cerebro evoluciona para descubrir una logica que esta en el mundo?
The zero derivative property of peaks and troughs underlies some o the most practical application of calculus. It allows us to use derivatives to figure out where a function reaches its maximum or minumum, best or cheapest way to do something.
Differential calculus it’s a certain specific compromise between the two paths considered.
Pythagorean theorem, as well as distance equals rate times time.
Snell’s law: describes how light rays bend when they pass from air into water, as they do when the sun shines into a swimming pool
Light moves more slowly in water, and it bends accordingly to minimize its travel time.
The eerie point is that light behaves as if it were considering all possible paths and then taking the best one. Nature somehow knows calculus.
Cuando un limite tiende a dos cosas, no existe. Tiene que tender a una cosa.
Chapter 18 Slices it Dices
Integral calculus is needed to sum all those changing forces.
The strength of gravity changes with distance- the closer two things are, the more strongly they attract.
Circadian rhytms within us: recurring naturally of 24 hour cycle, even in the absence of light fluctuations.
Algebra: changes steadily, constant rates
Calculus: changing rate.
Fundamental theorem of calculus: if you integrate the derivative of a function from one point to another, you get the net change in the function between the two points.
The strength of gravity changes with distance- the closer two things are, the more strongly they attract.
Circadian rhytms within us: recurring naturally of 24 hour cycle, even in the absence of light fluctuations.
Algebra: changes steadily, constant rates
Calculus: changing rate.
Fundamental theorem of calculus: if you integrate the derivative of a function from one point to another, you get the net change in the function between the two points.
Chapter 19 All about e
Pi: 3.14159…
i: it number of algebra, the imaginary number
e: exponential growth. 2.71828. e equals the limiting number approached by the sum as we take more and more terms.
i: it number of algebra, the imaginary number
e: exponential growth. 2.71828. e equals the limiting number approached by the sum as we take more and more terms.
Chapter 20 Loves me, loves me not
Inanimate things take the form of differential equations, which describe how intelinked variables change from moment to moment, depending on their current values.
Laws of physics are always expressed in the language of differential equations
Laws of physics are always expressed in the language of differential equations
Chapter 21 Step into the LIGHT
Vector Calculus, the branch of math that describes the invisible fileds all around us.
Maxwell discovered what light is
Vector: vehere: to carry, carrier of pathogen, is a step that carries you from one place to another.
Vectors: direction and magnitude
Velocities and forces.
The curl equations describe how the electric and magnetic fields interact and change over time. In so doing, these equations express a beautiful symmetry: they link one field’s rate of change in time to the other field’s rate of change in space, as quantified by its curl.
His symbol shuffling led him to the conclusion that electric and magnetic fields could propagate as a wave, somewhat like a ripple on a pond, except that these two fields were more like symbiotic organisms. Each sustained the other. The electric field’s undulations re-created the magnetic field, which in turn re-created the electric field, and so on, with each pulling the other forward, something neither could do on its own.
193,000 miles per second.
Maxwell discovered what light is
Vector: vehere: to carry, carrier of pathogen, is a step that carries you from one place to another.
Vectors: direction and magnitude
Velocities and forces.
The curl equations describe how the electric and magnetic fields interact and change over time. In so doing, these equations express a beautiful symmetry: they link one field’s rate of change in time to the other field’s rate of change in space, as quantified by its curl.
His symbol shuffling led him to the conclusion that electric and magnetic fields could propagate as a wave, somewhat like a ripple on a pond, except that these two fields were more like symbiotic organisms. Each sustained the other. The electric field’s undulations re-created the magnetic field, which in turn re-created the electric field, and so on, with each pulling the other forward, something neither could do on its own.
193,000 miles per second.
Chapter 22 The New Normal
The normal distribution can be proven to arise whenever a large number of mildly random effects of similar size, all a cting independently, are added together.
Power law distributions:
Power law distributions:
Chapter 23 Chances Are
Conditional probability- the probability that some event A happens, given the occurrence of some other event B.
The trick is to think in terms of natural frequencies- simple counts of events- rather than in more abstract notions of percentages, odds, or probabilities.
The trick is to think in terms of natural frequencies- simple counts of events- rather than in more abstract notions of percentages, odds, or probabilities.
Chapter 24 Untangling the Web
The algorithm starts with a guess, then updares all the pageranks by apportioning the fluid in equal shares to the outgoing links, and it keeps doing that in a series of rounds until everything settles down and all the pages get their rightful shares.
Chapter 25 The Loneliest Numbers
One is the loneliest number, and two can be as bad as one.
Number theory provides the basis for the encryption algorithms used millions of times each day to secure credit card transactions over the internet and to encode military-strenght secret communications.
They are fundamental, uncuttable, individible.
Definition of prime numbers to give us the theorem we want.
1 is only divisible only by 1 and itself.
Solitude of prime numbers: they fade into near oblivion.
The validity of the InN formula as N tends to infinity is now known as the prime number theorem.
Number theory provides the basis for the encryption algorithms used millions of times each day to secure credit card transactions over the internet and to encode military-strenght secret communications.
They are fundamental, uncuttable, individible.
Definition of prime numbers to give us the theorem we want.
1 is only divisible only by 1 and itself.
Solitude of prime numbers: they fade into near oblivion.
The validity of the InN formula as N tends to infinity is now known as the prime number theorem.
Chapter 26 Group Think
Group theory is necessarily abstract. It distills symmetry to its essence
They look for all the transformations that leave a shape unchanged, given certain constraints. These transformations are called the symmetries of the shape. Taken together they form a group, a collection of transformations whose relationships define the shape’s most basic architecture.
Vhart videos on youtube
They look for all the transformations that leave a shape unchanged, given certain constraints. These transformations are called the symmetries of the shape. Taken together they form a group, a collection of transformations whose relationships define the shape’s most basic architecture.
Vhart videos on youtube
Chapter 27 Twist and Shout
The niftiest application of topology is one that doesn’t involve Mobius strips at all. It’s a variation of the theme of twists and links.
Chapter 28 Think Globally
Like straight lines in ordinary space, great circles on a sphere contain the shortest paths between any two points. They’re called great because they’re the largest circles you have on a sphere.
Another property that lines and great circles share is that they’re the straightest paths between two points.
Great circles don’t do any additional curving above and beyond what they’re forced to do by following the surface of the sphere.
Einstein showed that light beams follow geodesics as they sail through the universe.
Light travels on geodesics through curved space-time, with the warping being caused by the sun’s gravity.
Keep it simple. But battling obstacles can give rise to great beauty- so much so that in art, and in math, it’s often more fruitful to impose constraints on ourselves.
Two points. Many paths, Mathematical bliss.
Another property that lines and great circles share is that they’re the straightest paths between two points.
Great circles don’t do any additional curving above and beyond what they’re forced to do by following the surface of the sphere.
Einstein showed that light beams follow geodesics as they sail through the universe.
Light travels on geodesics through curved space-time, with the warping being caused by the sun’s gravity.
Keep it simple. But battling obstacles can give rise to great beauty- so much so that in art, and in math, it’s often more fruitful to impose constraints on ourselves.
Two points. Many paths, Mathematical bliss.
Chapter 29 Analyze this
The associative law of addition, grouping the terms however we like, without affecting the ultimate answer.
Merely by adding its term in a different order, you can change the answer- something that could never happen for a finite sum. So even though the original series converges, it’s still capable of weirdness unimaginable in ordinary arithmetic.
When the basic building blocks are sine waves, the technique is known as Fourier analysis, and the corresponding sums are called Fourier series.
Gibbs phenomenon is the unwanted oscillations that cause blurring, shimmering, and other artifacts at sharp edges in the image.
Merely by adding its term in a different order, you can change the answer- something that could never happen for a finite sum. So even though the original series converges, it’s still capable of weirdness unimaginable in ordinary arithmetic.
When the basic building blocks are sine waves, the technique is known as Fourier analysis, and the corresponding sums are called Fourier series.
Gibbs phenomenon is the unwanted oscillations that cause blurring, shimmering, and other artifacts at sharp edges in the image.
Chapter 30 The Hilbert Hotel
Every real number between 0 and 1 is supposed to appear somewhere, at some finite place on the roster.
Hilbert Hotel can’t accommodate all the real numbers. There are simply too many of them, an infinity beyond infinity.
Hilbert Hotel can’t accommodate all the real numbers. There are simply too many of them, an infinity beyond infinity.