February 19, 2019 - Missing the Forest for the Tree: A Worldview Grounded in Science

820s: Algebra

Posted on February 19, 2019May 2, 2020 by Dan

Algebra is the study of mathematics by using a combination of symbols and values and the rules for manipulating those symbols and values. In its most basic form, it involves using equations to find the unknown. Linear equations, the quadratic formula, functions, and much more are all familiar examples of algebra. Algebra became recognized as a separate branch of mathematics thanks to work of the Persian mathematician Muhammad ibn Musa al-Khwarizmi.

The Roots of Algebra

The Quadratic Formula with Examples
(Credit: www.onlinemathlearning.com)

This history of any field of mathematics rides on a curvy road. This is no-less true for algebra. The roots of algebra can be traced back to Babylonian and Greek mathematics, at least 2000 BCE. We have evidence of stone tablets from Babylonian mathematicians who were hitting on the same ideas of algebra. The representation was not the same and the symbols they used were unique to their culture, but the fundamental spirit of algebra is evident. The Babylonians used complex arithmetic methods to solve modern algebraic problems. The Egyptians also worked with algebraic ideas but they were much less advanced that the Babylonians and did not advance much past solving linear equations.

The next big reservoir of algebraic thought came courteous of the Greek mathematicians, in particular a person named Diophantus of Alexandria. Diophantus lived in Alexandria, Egpyt in the 3rd century. Little is known of his life except his works and his age. He authored a thirteen book series titled Arithmetica that unfortunately has not survived in its full form. The portions that have survived show algebraic equations being solved. Diophantus and the Greeks devised a system of geometric algebra, using squares to solve for equations.

The Indian mathematician Brahmagupta was another person who influenced the development of algebra. Brahmagupta lived during the 7th century in northwest India. He wrote many influential works with a focus on mathematics and astronomy. His most famous work, Brahmasphutasiddhanta, provided solutions to linear and quadratic equations and is one of the earliest known texts to treat zero as a number. Much of his work moved from India the the Middle East, and was not known by Western Europe until many centuries later.

The Compendious Book on Calculation by Completion and Balancing

These earlier systems, especially the Greek and Indian, provided the inspiration for Persian mathematician al-Khwarizmi. Al-Khwarizmi was born in 780 and fortunate enough to have studied and worked in the House of Wisdom in Baghdad. The House of Wisdom was an enormous library and a major intellectual center of the time. In the 820s he wrote The Compendious Book on Calculation by Completion and Balancing, forming the foundation of algebra and establishing it as an independent discipline from arithmetic and geometry.

The Arabic title of his work is Al-jabr wa’l muqabalah, and it is from “al-jabr” that we get the term algebra. As the title indicates the text stresses the completion and balancing of equations. Here is a simple example of each type of operation:

Completion – Take the equation x+6=36. To complete this equation, we subtract 6 from each side to get x=36-6, or x=30.
Balancing – Take the equation x+y=y+30. To balance this equation we cancel y from both sides and get x=30.

His treatise is important because it presented the first systematic solution of linear and quadratic equations.

Algebra Today

Today algebra is used in a variety of mathematical fields, practical applications, and everyday life situations. Numbers and equations are used in everyday life whether we realize it or not. We use it in finance when we calculate loan interest, our return on investment, or a currency exchange rate. We use algebra when calculating rations. Ratios are relationships between different quantities. Twice as many guests are now showing up to your party? We need to balance the equation and add twice as many ingredients to that soup we are cooking. Are you a United States resident traveling outside the country? Most of the world uses the metric system for measurements and we’d use algebra to convert these measurements. We use algebra in statistics, graphing, computer coding, measuring calculations such as area, volume, and mass, and more.

Intuitively, we use algebra all the time when we solve for unknown variables. Abstractly, algebra helps us with our critical thinking and problem solving skills. Lastly many other branches of mathematics are dependent on algebra. Finding the area under a curve requires the use of calculus, and calculus would not be possible without algebra.

Continue reading more about the exciting history of science!

1848: Absolute Zero

Posted on February 19, 2019November 3, 2023 by Dan

Everyone is familiar with the concept of temperature. Temperature is a way to describe how hot or cold something is. But what is it that determines how hot or cold something is?

All of matter is made of atoms, and those atoms are always moving. Temperature then, is a measure of the kinetic energy (the energy of motion) of the particles in a substance or system. The faster the atoms move, the higher the temperature. Temperature also determines the direction of heat transfer, which is always from objects of a higher temperature to objects of a lower temperature.

Absolute zero is the lowest temperature theoretically possible. It corresponds to a bone-chilling -459.67 degrees on the Fahrenheit scale and -273.15 on the Celsius scale. At this temperature there is the complete absence of thermal energy, as the particles of a substance have no kinetic energy.

The History of Absolute Zero

The roots of the idea of absolute zero can be traced back to the early 17th century when scientists began to explore the behavior of gases. In 1665, Robert Boyle formulated Boyle’s Law, which stated that the volume of a gas is inversely proportional to its pressure at a constant temperature. This law laid the foundation for the study of gases and eventually lead to the concept of absolute zero.

Over the next 200 years additional discoveries were made that brought scientists closer and closer to the concept of an absolute zero temperature point. Then in 1848 the distinguished British scientist, William Thomson (later Lord Kelvin), published a paper titled On an Absolute Thermometric Scale where he made the case for a new temperature scale with the lower limit to be absolute zero. At this time temperature was measured on various scales, such as Celsius and Fahrenheit scales. However these scales had certain limitations and were based on arbitrary reference points. Thomson recognized the need for a temperature scale that would provide a universal standard and be based on fundamental physical principles. Scientists could now rely on a scale for temperature measurements without the need for using negative numbers.

Thomson’s key insight was to base his new scale on the behavior of an ideal gas. According to Boyle’s Law, the pressure of an ideal gas is directly proportional to its temperature when the volume is held constant. He realized that if a gas were to be cooled to a temperature at which its volume reached zero, then this temperature would represent the absolute zero of temperature. Thomson correctly calculated its value and used Celsius as the scale’s unit increment.

Absolute zero is the temperature to which you all atoms would stop moving and kinetic energy equals zero. This temperature has never been achieved in the laboratory, but it’s been close. Sophisticated technology involving laser beams to trap clouds of atoms held together by magnetic fields generated by coils have cooled elements such as helium to within fractions of a degree of absolute zero. The current world record for the coldest temperature is held by a team of researchers at Standford University in 2015. They used sophisticated laser beams to slow rubidium atoms, cooling them to an incredible 50 trillionth of a degree, or 0.00000000005 degrees Celsius, above absolute zero! This is extremely impressive since according to theory, it is suggested that we will never be able to achieve absolute zero.

Thomson’s temperature scale was later named the Kelvin Scale in his honor, and kelvin is the International System of Units (SI) base unit of temperature.

Practical Uses of Absolute Zero

The concept of absolute zero is relevant to many modern technologies, such as cryogenics and quantum computing. Below is a summary of its applications:

Cryogenics – cryogenics is the study of very low temperatures. cryogenics is the study of very low temperatures. Its technologies are used in various industries such as medical science, where they assist in the preservation and storage of biological materials.
Superconductivity – superconductivity is the phenomenon where certain materials can conduct electric current with zero electrical resistance. Superconductivity is needed in several fields including medical imaging (MRI) and particle accelerator technologies.
Quantum Computing – at very low temperatures quantum mechanical effects become more pronounced.at very low temperatures quantum mechanical effects become more pronounced. In order to create and manipulate qubits, the basic unit of quantum information, quantum computing systems require extremely low temperatures.
Space Exploration – extremely low temperatures are encountered in deep space. Understanding the properties of materials at these temperatures is crucial for designing spacecraft components.

As you can see, absolute zero holds profound implications for various fields of study and cutting-edge technology.

Continue reading more about the exciting history of science!

Henry Cavendish

Posted on February 19, 2019January 23, 2024 by Dan

Henry Cavendish (1731 – 1810) was one of the great experimental and theoretical chemist and physicist of the 18th century. So meaningful were his contributions to science that James Clerk Maxwell named the University of Cambridge’s physics laboratory in his honor after he founded it in 1874.

Henry was born in Nice, France, due to his family traveling at the time of his birth. He was educated in a private school in London then attended the University of Cambridge in 1748 where he stayed for three more years. His father, Lord Charles Cavendish, was involved with members of the Royal Society of London and took Henry to meetings in the last 1750s. By 1760 Henry was became an elected member of the Royal Society and from there on lived a life dedicated to science.

His interest and achievements in science were vast and wide ranging. He began his work at the Royal Society by heading a committee to review the society’s meteorological instruments. This initiated his research in chemistry and in particular gas chemistry. His is credited with being the first person to isolate hydrogen (which he termed “inflammable air”), to correctly calculate its density, and determine that it was contained in water in a two to one proportion. As with most scientists of his time, Henry also experimented with electricity. He wrote many papers on electricity for the Royal Society but most of his experiments did not become known until many years after his death.

He was known for his extremely careful and accurate measurements. This quality came in handy when it came time for him to measure the composition of the atmosphere and the density of the Earth. Both measurements he obtained compare very nicely with the values accepted today.

Henry Cavendish amassed incredible wealthy throughout his life. He used his wealthy mainly in the pursuit of science as he was not very sociable. It is thought that he had Asperger syndrome, a form of autism. He died in 1810 as one of the wealthiest men in Britain.