![\"\"](\"https:\/\/world.edu\/wp-content\/uploads\/2024\/10\/mathematical-history.jpg\")
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On October 16 1843, the Irish mathematician William Rowan Hamilton had an epiphany during a walk alongside Dublin\u2019s Royal Canal. He was so excited he took out his penknife and carved his discovery right then and there on Broome Bridge.<\/p>\n
It is the most famous graffiti in mathematical history, but it looks rather unassuming:<\/p>\n
Yet Hamilton\u2019s revelation changed the way mathematicians represent information. And this, in turn, made myriad technical applications simpler \u2013 from calculating forces when designing a bridge, an MRI machine or a wind turbine, to programming search engines and orienting a rover on Mars. So, what does this famous graffiti mean?<\/p>\n
The mathematical problem Hamilton was trying to solve was how to represent the relationship between different directions in three-dimensional space. Direction is important in describing forces and velocities, but Hamilton was also interested in 3D rotations.<\/p>\n
Mathematicians already knew how to represent the position of an object with coordinates such as\u00a0x<\/em>,\u00a0y<\/em>\u00a0and\u00a0z<\/em>, but figuring out what happened to these coordinates when you rotated the object required complicated spherical geometry. Hamilton wanted a simpler method.<\/p>\n He was inspired by a remarkable way of representing two-dimensional rotations. The trick was to use what are called \u201ccomplex numbers\u201d, which have a \u201creal\u201d part and an \u201cimaginary\u201d part. The imaginary part is a multiple of the number\u00a0i<\/em>, \u201cthe square root of minus one\u201d, which is defined by the equation\u00a0i<\/em>\u00a0\u00b2\u00a0=\u00a0\u20131.<\/p>\n By the early 1800s several mathematicians, including Jean Argand and John Warren, had discovered that a complex number can be represented by a point on a plane. Warren had also shown it was mathematically quite simple to rotate a line through 90\u00b0 in this new complex plane, like turning a clock hand back from 12.15pm to 12 noon. For this is what happens when you multiply a number by\u00a0i<\/em>.<\/p>\n When a complex number is represented as a point on a plane, multiplying the number by i amounts to rotating the corresponding line by 90\u00b0 anticlockwise.<\/p><\/div>\n Hamilton was mightily impressed by this connection between complex numbers and geometry, and set about trying to do it in three dimensions. He imagined a 3D complex plane, with a second imaginary axis in the direction of a second imaginary number\u00a0j<\/em>, perpendicular to the other two axes.<\/p>\n It took him many arduous months to realise that if he wanted to extend the 2D rotational wizardry of multiplication by\u00a0i<\/em>\u00a0he needed\u00a0four<\/em>-dimensional complex numbers, with a\u00a0third<\/em>\u00a0imaginary number,\u00a0k<\/em>.<\/p>\n In this 4D mathematical space, the\u00a0k<\/em>-axis would be perpendicular to the other three. Not only would\u00a0k<\/em>\u00a0be defined by\u00a0k<\/em>\u00a0\u00b2\u00a0=\u00a0\u20131, its definition also needed\u00a0k<\/em>\u00a0=\u00a0ij<\/em>\u00a0=\u00a0\u2013ji<\/em>. (Combining these two equations for\u00a0k<\/em>\u00a0gives\u00a0ijk<\/em>\u00a0=\u00a0\u20131.)<\/p>\n Putting all this together gives\u00a0i<\/em>\u00a0\u00b2\u00a0=\u00a0j<\/em>\u00a0\u00b2\u00a0=\u00a0k<\/em>\u00a0\u00b2\u00a0=\u00a0ijk<\/em>\u00a0=\u00a0\u20131, the revelation that hit Hamilton like a bolt of lightning at Broome Bridge.<\/p>\n Hamilton called his 4D numbers \u201cquaternions\u201d, and he used them to calculate geometrical rotations in 3D space. This is the kind of rotation used today to move a robot, say, or orient a satellite.<\/p>\n But most of the practical magic comes into it when you consider just the imaginary part of a quaternion. For this is what Hamilton named a \u201cvector\u201d.<\/p>\n A vector encodes two kinds of information at once, most famously the magnitude and direction of a spatial quantity such as force, velocity or relative position. For instance, to represent an object\u2019s position (x<\/em>,\u00a0y<\/em>,\u00a0z<\/em>) relative to the \u201corigin\u201d (the zero point of the position axes), Hamilton visualised an arrow pointing from the origin to the object\u2019s location. The arrow represents the \u201cposition vector\u201d\u00a0x<\/em>\u00a0i<\/em>\u00a0+\u00a0y<\/em>\u00a0j<\/em>\u00a0+\u00a0z<\/em>\u00a0k<\/em>.<\/p>\n This vector\u2019s \u201ccomponents\u201d are the numbers\u00a0x<\/em>,\u00a0y<\/em>\u00a0and\u00a0z<\/em>\u00a0\u2013 the distance the arrow extends along each of the three axes. (Other vectors would have different components, depending on their magnitudes and units.)<\/p>\n A vector (r) is like an arrow from the point O to the point with coordinates (x, y, z).<\/p><\/div>\n Half a century later, the eccentric English telegrapher Oliver Heaviside helped inaugurate modern vector analysis by replacing Hamilton\u2019s imaginary framework\u00a0i<\/em>,\u00a0j<\/em>,\u00a0k<\/em>\u00a0with real unit vectors,\u00a0i<\/strong>,\u00a0j<\/strong>,\u00a0k<\/strong>. But either way, the vector\u2019s components stay the same \u2013 and therefore the arrow, and the basic rules for multiplying vectors, remain the same, too.<\/p>\n Hamilton defined two ways to multiply vectors together. One produces a number (this is today called the scalar or dot product), and the other produces a vector (known as the vector or cross product). These multiplications crop up today in a multitude of applications, such as the formula for the electromagnetic force that underpins all our electronic devices.<\/p>\n Unbeknown to Hamilton, the French mathematician Olinde Rodrigues had come up with a version of these products just three years earlier, in his own work on rotations. But to call Rodrigues\u2019 multiplications the products of vectors is hindsight. It is Hamilton who linked the separate components into a single quantity, the vector.<\/p>\n Everyone else, from Isaac Newton to Rodrigues, had no concept of a single mathematical object unifying the components of a position or a force. (Actually, there was one person who had a similar idea: a self-taught German mathematician named Hermann Grassmann, who independently invented a less transparent vectorial system at the same time as Hamilton.)<\/p>\n Hamilton also developed a compact notation to make his equations concise and elegant. He used a Greek letter to denote a quaternion or vector, but today, following Heaviside, it is common to use a boldface Latin letter.<\/p>\n This compact notation changed the way mathematicians represent physical quantities in 3D space.<\/p>\n Take, for example, one of Maxwell\u2019s equations relating the electric and magnetic fields:<\/p>\n With just a handful of symbols (we won\u2019t get into the physical meanings of \u2202\/\u2202t<\/em>\u00a0and \u2207\u00a0\u00d7), this shows how an electric field vector (E<\/strong>) spreads through space in response to changes in a magnetic field vector (B<\/strong>).<\/p>\n Without vector notation, this would be written as three separate equations (one for each component of\u00a0B<\/strong>\u00a0and\u00a0E<\/strong>) \u2013 each one a tangle of coordinates, multiplications and subtractions.<\/p>\n The expanded form of the equation. As you can see, vector notation makes life much simpler.<\/p><\/div>\n I chose one of Maxwell\u2019s equations as an example because the quirky Scot James Clerk Maxwell was the first major physicist to recognise the power of compact vector symbolism. Unfortunately, Hamilton didn\u2019t live to see Maxwell\u2019s endorsement. But he never gave up his belief in his new way of representing physical quantities.<\/p>\n Hamilton\u2019s perseverance in the face of mainstream rejection really moved me, when I was researching\u00a0my book on vectors<\/a>. He hoped that one day \u2013 \u201cnever mind when\u201d \u2013 he might be thanked for his discovery, but this was not vanity. It was excitement at the possible applications he envisaged.<\/p>\n A plaque on Dublin\u2019s Broome Bridge commemorate\u2019s Hamilton\u2019s flash of insight. Cone83 \/ Wikimedia,<\/p><\/div>\n He would be over the moon that vectors are so widely used today, and that they can represent digital as well as physical information. But he\u2019d be especially pleased that in programming rotations, quaternions are still often the best choice \u2013 as NASA and computer graphics programmers know.<\/p>\n In recognition of Hamilton\u2019s achievements, maths buffs\u00a0retrace his famous walk<\/a>\u00a0every October 16 to celebrate Hamilton Day. But we all use the technological fruits of that unassuming graffiti every single day.<\/p>\n![\"\"](\"https:\/\/world.edu\/wp-content\/uploads\/2024\/10\/XRealNumbers.jpg\")
Quaternions and vectors<\/strong><\/h4>\n
![\"\"](\"https:\/\/world.edu\/wp-content\/uploads\/2024\/10\/vectors.jpg\")
A single mathematical object<\/strong><\/h4>\n
\n\u2207 \u00d7 E = \u2013\u2202 B \/\u2202t<\/div>\n<\/div>\n![\"\"](\"https:\/\/world.edu\/wp-content\/uploads\/2024\/10\/ExpandedForm.jpg\")
The power of perseverance<\/strong><\/h4>\n
![\"\"](\"https:\/\/world.edu\/wp-content\/uploads\/2024\/10\/Plaque.jpg\")