{"id":388,"date":"2023-12-04T07:53:24","date_gmt":"2023-12-04T07:53:24","guid":{"rendered":"http:\/\/bullseye.ac\/blog\/?p=388"},"modified":"2023-12-04T07:57:15","modified_gmt":"2023-12-04T07:57:15","slug":"the-joy-of-x-by-steven-strogatz","status":"publish","type":"post","link":"https:\/\/bullseye.ac\/blog\/book-reviews-summary\/the-joy-of-x-by-steven-strogatz\/","title":{"rendered":"‘The Joy of X’ by Steven Strogatz"},"content":{"rendered":"\n
\u2026 A summary by Vinay Wagh<\/em><\/em><\/h5>\n\n\n\n

We carry a common myth that most of the Math that is being taught in the schools and colleges is useless. But that\u2019s not true.  Math was formulated to satisfy the needs of previous centuries and is now used to handle the modern technologies. The way math was formulated and  is being  used is meticulously answered   in this book by Steven in an interesting way that truly connects practical life<\/p>\n\n\n\n

Why sum of the n odd natural numbers is n2<\/sup>?<\/p>\n\n\n\n

1 + 3 = 22<\/sup>, 1 + 3 + 5 =32<\/sup> and 1+ 3 + 5 = 42<\/sup>\u2026and so on<\/p>\n\n\n\n

Why is the sum of n natural numbers n(n+1)\/2? <\/p>\n\n\n\n

1+2 +3 + 4 is?<\/p>\n\n\n\n

Put white marbles on the floor in the given order. 1, then 2, then 3 and then 4. Now make a rectangle by adding black marbles.  Now it becomes a rectangle of four rows and five columns. (4 x 5). There are 20 marbles. So how many are white? Half of 20 = 10. Area of triangle is half of the area of rectangle. So n(n+1)\/2<\/p>\n\n\n\n

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The enemy of my enemy: Negativity multiplied by negativity turns it into positivity.<\/strong><\/p>\n\n\n\n

 Scholars have used these ideas to analyze the run-up to world war-I. It shows the shifting alliances from Great Britain, France, Russia, Italy, Germany, Austria and Hungary between 1872 and 1907. The bold segment between the countries shows friendly relationship.  The dotted segment indicates Hostile relationship.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n
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\"\"\/<\/figure>\n\n\n\n

The bold line represents friendly relation whereas the dotted line indicates enmity <\/p>\n\n\n\n

Some part of what we observe is due to nothing more than the primitive logic of \u201cThe enemy of my enemy\u201d and this part is captured perfectly by the multiplication of negative numbers.<\/p>\n\n\n\n

Commutative law application: <\/strong><\/p>\n\n\n\n

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  1. 10 % Tax to be applied after a discount of 20%<\/li>\n\n\n\n
  2. 20% discount to be applied on a value that is increased by 10% tax. <\/li>\n<\/ol>\n\n\n\n

    Although it is the same, quite a few find it different. You will end getting 12% decrease on the given marked price. All it says is a x b = b x a<\/p>\n\n\n\n

    The commutative law does not always work with our mindsets. Let\u2019s say that a person wants to get into IIM-B very desperately. Not getting there can lead her to take a silly step of committing suicide. The person gets rejected by IIM-B but gets a call from IIM-C. The girl cancels the idea of committing suicide. She can go to IIM-C and commit suicide, but not the other way round.<\/p>\n\n\n\n

    Early in the development of quantum mechanics, Werner Heisenberg and Paul Dirac had discovered that nature follows a curious kind of logic in which p x q \u2260 q x p where p and q represent the momentum and position of quantum particle. Without that break-down of commutative law, there would be no Heisenberg uncertainty principle, atoms would collapse and nothing would exist. That\u2019s why you better mind your p\u2019s and q\u2019s. <\/p>\n\n\n\n

    Division and its Discontents:<\/strong><\/p>\n\n\n\n

    0.12122122212222\u2026..<\/p>\n\n\n\n

    By design, the blocks of 2 get progressively longer as we move to the right. There is no way to express this decimal fraction. It\u2019s irrational!<\/p>\n\n\n\n

    The joy of x<\/strong><\/p>\n\n\n\n

    Grant wiggins, an education consultant, has been posing a problem to students and faculties for years. More than 50% people got it wrong.<\/p>\n\n\n\n

    Try and make an equation for a relation, \u201cOne yard equals three feet\u201d. <\/p>\n\n\n\n

    If you think the answer is 1 y = 3f, welcome to the club of 50%. Seems to be a straightforward translation but as soon as you try a few numbers, you\u2019ll see that this formula does not work. Say the hallway is 10 yards long. We all know it is 30 feet. Now plug y = 10 and f =30 the formula does not work. <\/p>\n\n\n\n

    The correct equation is f = 3y. Here 3 really means \u201c3 feet per yard\u201d. When you multiply it by Y in yards, the units of yards cancel out and you\u2019re left with units of feet, as you should be.<\/p>\n\n\n\n

    Imagine imaginary numbers<\/strong><\/p>\n\n\n\n

    What multiplying by i (imaginary number) <\/em>looks like?<\/p>\n\n\n\n

    \"\"\/<\/figure>\n\n\n\n

    Multiplying by \u2018I\u2019 produces a rotation counterclockwise by a quarter turn. It takes an arrow of length 2 pointing east and changes it into a new arrow of the same length but now pointing north. Electrical engineers love complex numbers for exactly this reason. Having such a compact way to represent 90o<\/sup> rotations is very useful when working with Alternating currents and magnetic fields, because these often involve oscillations or waves that are quarter cycles. In aerospace engineering they eased the first calculations of the lift on an aero plane wing. Civil and mechanical engineers use them routinely to analyze the vibrations of footbridge, skyscrapers and cars driving on bumpy roads.<\/p>\n\n\n\n

    The 90o<\/sup> rotation property also sheds light on what i2<\/sup> = -1 really means. If we multiply positive number by i2<\/sup>, the corresponding arrow rotates 180o<\/sup>, flipping the arrow from east to west.<\/p>\n\n\n\n

    \"\"\/<\/figure>\n\n\n\n

    Why the name Algebra?<\/p>\n\n\n\n

    In the early part of ninth century, Muhammad bin Musa al-Khwarizmi, a mathematician working in Baghdad, wrote a seminal text book in which he highlighted the usefulness of restoring a quantity being subtracted by being added to the other side of equation. Say x + 2 = 4. We subtract 2 on both sides to get x +2 -2 = 4 -2. So x = 2. He called this process as Al-jabr (Arabic word for \u2018restoring\u2019) which later morphed into \u2018Algebra\u201d. Then long after his death, he hit the etymological jackpot again. His own name, Al- khwarizmi, lives on today in word, \u201cAlgorithm\u201d.<\/p>\n\n\n\n

    Quadratic Equations<\/strong><\/p>\n\n\n\n

    X2<\/sup> + 10 x = 39<\/p>\n\n\n\n

    Al-Khwarizmi put the puzzle this way X2<\/sup> + 10x = 39<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    The missing corner was an irresistible step. That was filled in to turn the figure into a perfect square. <\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    Power tools<\/strong><\/p>\n\n\n\n

    A mathematician needs Functions for the same reason that a builder needs hammers and drills. Tools transform things. So does functions. <\/p>\n\n\n\n

    If we plot y = 4 \u2013x2<\/sup> on graph we get <\/p>\n\n\n\n

    \"\"\/<\/figure>\n\n\n\n

    The bowed shape of the curve is due to the action of mathematical pliers. The function that transforms x into x2<\/sup> behaves a lot like the common tool for bending and pulling things. And what role does the 4 play in the equation? It acts like a nail for hanging a picture on a wall. It lifts the bent wire arch by 4 units. Since it raises all the points by the same amount it\u2019s known as a constant function. The tool x2<\/sup> bends the piece of X-axis and the \u20184\u2019 lifts it and holds it. These building blocks 4 and -x2<\/sup> can be regarded as component parts of a more complicated function, 4- x2<\/sup>, just as wires, batteries and transistors are components of a radio. If you look this way, you\u2019ll start noticing functions everywhere.<\/p>\n\n\n\n

    Power functions of the form xn<\/sup> in which the variable x is raised to a fixed power n. changing n yields handy tools. For parabola n = 2 and for constant n = 0. Whereas n =1 gives a function that works like a ramp, a steady incline or a decline. It\u2019s called linear function as it produces a line on the graph. What it n = -2? (1\/x2<\/sup>). This function is good to describe how waves and forces attenuate as they spread out in three dimensions the way sound softens as it moves away from the source. <\/p>\n\n\n\n

    A Paper can be folded only seven to eight times.<\/strong><\/p>\n\n\n\n

    In x2<\/sup>, as x gets larger and larger an exponential function of x eventually grows faster than any power function no matter how large the power. Exponential growth in almost unimaginably rapid.  That\u2019s why it\u2019s so hard to fold a piece of paper in half more than seven or eight times. Each folding approximately doubles the thickness of the wad, causing it to grow exponentially. Meanwhile the Wad\u2019s length shrinks in half every time and thus decreases exponentially fast. For standard sheet of the note book paper, the wad becomes thicker than it is long and so it can\u2019t be folded again. For a sheet to be considered legitimately folded n times, the resulting wad is required to have 2n<\/sup> layers in a straight line, and this can\u2019t happen if the wad is thicker than it is long. In 2002 Britney Gallivan, then a junior in high school solved. She made a formula that predicted the maximum number of times, n, that paper of a given thickness T and length L could be folded in one direction.<\/p>\n\n\n\n

    L = T6 (2n<\/sup> + 4)( 2n<\/sup> \u2013 1). <\/p>\n\n\n\n

    The conic Conspiracy:<\/strong><\/p>\n\n\n\n

    Whispering galleries are remarkable acoustic spaces found beneath certain domes, vaults or curved ceilings. There is famous whispering gallery in New York, the Grand central station. It\u2019s a fun place to take a date. You can communicate by whispering when you are forty feet apart and separated by a bustling passageway. You\u2019ll hear each other clearly but the passersby won\u2019t hear a word you\u2019re saying. <\/p>\n\n\n\n

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