Author Archives: arithmomancy

Transcendental Numbers: Beyond Algebra

The history of mathematics has been fraught with disappointments for mathematicians. This is particularly true in regard to the expected and continued failure of numbers, and math in general, to be pure and graceful. The Pythagoreans, in 6th century BCE Greece, venerated the whole numbers with an almost religious devotion because of their purity, and believed that the universe could be described by using only whole numbers. Unfortunately, math is not as pure as the Pythagoreans thought, which was revealed first by Hippasus of Metapontum when he discovered an undeniable proof for the existence of irrational numbers. Incidentally, Pythagoras had poor Hippasus drowned because of this. (The tale of the drowning of Hippasus may be merely a legend, like much of what is “known” about the Pythagoreans, due to a lack of reliable sources from the period.) Another impurity of numbers was wrestled with for millennia in the form of the square roots of negative numbers, a problem that was only put to rest with the advent of the imaginary number i. However, numbers turned out to be even weirder than previously imagined, because transcendental numbers were discovered by Gottfried Wilhelm Leibniz, in 1682.

In order to understand transcendental numbers, we need to understand algebraic numbers, or numbers that are not transcendental. An algebraic number is any number that is the solution to a polynomial with rational coefficients. Rational numbers are numbers that can be written as the ratio of two integers. All rational numbers are algebraic numbers, for instance the number 2 is a rational number because it can be written as 2/1. It is also an algebraic number because it is the root of the polynomial X – 2 = 0, which is a polynomial with only rational coefficients. While all rational numbers are algebraic, not all are algebraic numbers are rational, for example, √ (2) is an irrational number, but it is also algebraic because it is the solution to X² – 2 = 0. Strangely, the imaginary number i, although it is not real, is an algebraic number since it is the root of the polynomial X² + 1 = 0.

Transcendental numbers are numbers that cannot be written as the root to a polynomial with rational coefficients. All transcendental numbers are irrational. Leibniz coined the term “transcendental” in his 1682 paper in which he proved that the sin function is not an algebraic function.  Leonhard Euler (1707- 1783) was the first to generally define transcendental numbers in the modern sense, although it was Joseph Liouville, in 1844, who definitively proved the existence of the first transcendental number. That number is now called the Liouville Constant, and it is .110001000… with a 1 in every n! place after the decimal. The Liouville Constant was specifically constructed by Liouville to be a transcendental number. However, Charles Hermite first identified a transcendental number that was not created for that purpose in 1873. That number was the constant e, or Euler’s number, and is the base of the natural logarithm.

A famous transcendental number, called “Champernowne’s Number,” was discovered in 1933 and named after David G. Champernowne. It is formed by concatenating all the natural numbers behind the decimal point 0.12345678910…. Although, easily the most famous transcendental number is pi, which was proved to be transcendental by Ferdinand von Lindemann in 1882.

Pi. Image: Travis Morgan, via Flickr.

Pi. Image: Travis Morgan, via Flickr.

Georg Cantor, in the1870’s, proved that there are as many transcendental numbers as real numbers, a concept that is mind-boggling since the real numbers are uncountable. However, only a few numbers have ever been definitively proven to be transcendental, because it is extremely difficult to prove that any given number is transcendental.

Along with irrational and imaginary numbers, transcendental numbers have challenged and frustrated mathematicians throughout the ages. Undoubtedly, Pythagoras would be horrified by transcendental numbers, or maybe he would just drown anyone who tried to tell him about them. Today, however, transcendental numbers are embraced by mathematicians as a deep and important part of math.


Imaginary Numbers: From Outcast to Respectability

Image: Matheepan Panchalingam, via Flickr.

Image: Matheepan Panchalingam, via Flickr.

Imaginary numbers, which are also known as complex numbers, have had a pretty bad reputation. When most people think of imaginary numbers, they probably break out in a cold sweat from the horrific memories of high school math class. They think that imaginary numbers are utterly incomprehensible and useless in the “real” world. “Imaginary numbers” sound very intimidating to people who are not familiar with them. They also sound highly theoretical with little or no use outside of pure mathematics. In fact, the exact opposite is true.

The most common imaginary number is i, which is formally defined as i = √-1. Since the act of squaring any real number always makes the number positive– whether it began as a negative number or not, it is impossible to find the square root of a negative number without using i. Thus, i made possible an entire class of math problems that were not possible before. For example, √-64 = 8i, cannot be done without using i, because √-64 does not exist in the real number line. Additionally, i can be easily changed from an “imaginary” number into a “real” number simply by squaring it: i² = -1.

The first known person to stumble upon the idea of using an imaginary number to take the square root of a negative number was the Greek mathematician Heron of Alexandria in 50 CE. He was trying to find the volume of a section of a pyramid using a formula that involved the slant height of the pyramid. However, certain values for the slant height would produce the square root of a negative number. Heron was very uncomfortable with this result, so in order to avoid using a negative number, he fudged his calculation by dropping the negative sign.

Girolamo Cardano was an Italian mathematician who was particularly interested in finding the solutions to cubic and quartic equations. In 1545, he published a book titled Ars Magna, which contained the solutions to cubic and quartic equations. One of the equations in his book gave the solution of 5 ± √-15. Commenting on this equation, Cardano wrote, “Dismissing mental tortures, and multiplying 5 + √ – 15 by 5 – √-15, we obtain 25 – (-15). Therefore the product is 40. …. and thus far does arithmetical subtlety go, of which this, the extreme, is, as I have said, so subtle that it is useless.”

Perhaps the first champion of imaginary numbers was Italian mathematician, Rafael Bombelli (1526-1572). Bombelli understood thattimes should equal -1, and that -i times should equal one. However, Bombelli could not find a practical use for this property, so he generally was not believed. Bombelli did have what people called a “wild idea” – that imaginary numbers could be used to get real answers.

Imaginary numbers continued to live in disgrace until the work of a series of mathematicians in the 18th and 19th centuries. Leonhard Euler helped clear up some of the problems with using imaginary numbers by developing the notation i to mean √-1. He also introduced the notation a+bi for complex numbers. Carl Friedrich Gauss  made imaginary numbers much more concrete and less “imaginary” when he graphed imaginary numbers as points on the complex plane in 1799. However, William Rowan Hamilton in 1833, delivered the coup de grace to imaginary numbers’ bad name when he advanced the idea that complex numbers could be expressed as a pair of real numbers. For example 4+3i could be written simply as (4,3). This made complex numbers much easier to understand and use.

Today, imaginary numbers are an essential part of the everyday calculations that make modern technology work. They are indispensable in the field of electrical engineering, particularly in the analysis of alternating current, like the electrical current that powers household appliances. Also, cell phones and air travel would not be possible without imaginary numbers because they are necessary in the computations involved in signal processing and radar. Imaginary numbers are even used by biologists when studying the firing events of neurons in the brain. Imaginary numbers have come a long way in the five hundred years since they were scoffed at for being absurd and totally useless.


Egyptian Numbers: Counting By Ten

When most people think of early mathematics, the first thing that comes to mind is probably Archimedes or Pythagoras of ancient Greece; or perhaps ancient Mesopotamia and its thousands of clay tablets, many of them containing math problems.  Few people think of ancient Egypt. This is mainly due to the fact that little is known about ancient Egyptian math in comparison to these other civilizations. However, by looking at what we do know about Egyptian math and the feats of engineering they accomplished with it, we can recognize how complex and sophisticated it was.

Rhind Mathematical Papyrus

A fragment from the Rhind Mathematical Papyrus. Image: Public domain, via Wikimedia Commons.

The majority of our knowledge of ancient Egyptian math comes from two papyri: the Rhind Mathematical Papyrus, which was originally written about 1985-1975 BCE; and the Moscow Mathematical papyrus, dating from around 1850 BCE. Math problems were not written on the walls of temples or great monuments in Egypt, so all math was probably done on papyrus or other perishable media, meaning that many mathematical works have not come down to us. This in in contrast to the mathematical documents from Mesopotamia, which were primarily done on clay tablets that are not as susceptible to the elements and ravages of time.

The ancient Egyptians probably developed the first base ten numerical system in human history. It was fully in use around 2700 BCE and perhaps even earlier. They had different symbols representing the powers of ten. A straight vertical line represented one, 10 was shown by a drawing of a hobble for cattle, a picture of a coil of rope served as a symbol for 100, a lotus plant delineated 1,000, a bent finger for 10,000, a tadpole or a frog for 100,000 and the picture of a god, perhaps Heh, represented 1 million. Although the Egyptian numerals were in a decimal system, it was not a positional place value system like the decimal system we use, or the sexagesimal (base 60) system developed by the Mesopotamians. Multiples of the powers of ten were written by repeating the symbol as many times as needed, and although they had a symbol for 1 million, other large numbers would have been very tedious to write. For instance, the number 987,654 would have required 39 characters. This Egyptian system possibly had an influence on the later Greek numerical system, but the Greeks improved upon it, creating a different symbol for each number 1-9, and other symbols for 20, 30 and so on.

Parts of the Eye of Horus were used to write fractions. Image: BenduKiwi, via Wikimedia Commons.

Parts of the Eye of Horus were used to write fractions. Image: BenduKiwi, via Wikimedia Commons.

The Egyptians also had notation for fractions, although all Egyptian fractions were unit fractions (meaning they always had one in the numerator), with only two exceptions, 2/3 and 3/4. Fractions were marked by the hieroglyph for ‘R’ which is a long skinny oval, very similar in shape to the Eye of Horus. In fact, in an ancient Egyptian myth, the evil god Seth attacked his brother the god Horus and in the fight Seth gouged out the eye of Horus and tore it into pieces, fortunately the god Throth was able to put Horus’s eye back together and heal it. Thus, to honor the gods and this myth, whenever the Egyptians used fractions in relation to their measurement of volume, which is the hekat, the commonly used fractional parts were transcribed by using the corresponding parts of the Eye of Horus.

The great feats of engineering that the ancient Egyptians accomplished would have required an enormous amount of sophisticated math. The pyramids, for instance, are considered a marvel of mathematics and engineering. The base of the pyramids are almost perfect squares which the Egyptians would have achieved by using trigonometry, like the 3-4-5 trick. It was known in Egypt that a triangle with 3, 4 and 5 unit sides would always be a perfect right triangle. This Pythagorean triple, observed by the Egyptians long before Pythagoras, is sometimes called an “Egyptian triangle.” The Egyptians would have utilized this rule of geometry when laying out the base of a pyramid by tying knots in rope at 3, 4 and 5 unit intervals. Other tricks of trigonometry were known as well. The Rhind Mathematical Papyrus contains an equation for calculating the slope of a pyramid’s face, which is the same thing as finding the cotangent. It is also likely that they knew how to find a pyramid’s volume.

It is unfortunate that more is not known about the mathematics that were used by the ancient Egyptians, because they were obviously very skillful and innovative, and they must have been among the first people to develop important mathematical principals.