Why there's a big need for big numbers

Changed caption to "Three Lake Forest College students were first place among Illinois teams at a problem solving competition in 2018, from left: Kaloyan Parvanov, '20; Emma Juettner, '20; Associate Professor of Mathematics Enrique Treviño, and Dipika Subramaniam, '21." at author's request.

"Why do people need big numbers?," asked a young Grayslake Area Public Library District patron.

Calculating speed and distance can require big numbers. Big numbers can measure big things, like the number of atoms in the universe or nano scale objects, such as subatomic bits on an electron.

Words define large numbers, such as googolplex for 10 to the power of a googol. Mathematicians have developed a few ways to reflect big numbers by using decimals and exponents. Logarithms, a type of shorthand for equations, make it easier for big numbers to be multiplied.

Ancient Greeks, such as Euclid, used big numbers for astronomy calculations. Euclid also wrote about mirror refractions, perspective and music theory - concepts that can require big numbers or nontraditional equations.

While Euclid's thorough examination of mathematics, including geometry, math theorems and proofs, has been taught for centuries, other civilizations developed numbering systems showing a need for large numbers. Mayans used a system with base 20 and introduced the concept of zero. In Ancient India, multiple numbering systems and big number exponents were calculated. Arabs in the eighth century developed a method for multiplication of decimals.

Enrique Treviño, associate professor of math at Lake Forest College, said there's always been a need for large numbers.

"I think large numbers come into play in societies for natural reasons. One could try to estimate the number of atoms in the universe or try to find out how many seconds have elapsed since the Big Bang or even try to count how many games of chess are possible," Treviño said. "Lots of simple counting exercises lead to huge numbers."

Charting a course to Mars requires a steady stream of numbers to calculate fuel needs. Another urgent need for big numbers today goes unseen - the numbers computers use to create simple content, such as emails.

"Nowadays, we also use large numbers to secure our internet conversations," Treviño said. "Namely, the algorithms for security often use the fact that multiplication is easy but factoring is hard. The use of very large primes with over 200 digits is common place in security applications."

Exponents give heft to smaller numbers. Treviño developed a theorem that includes numbers raised to the power of 4732. His students' work, published recently in the math journal "Elemente der Mathematik", assigned numbers to letters of the alphabet and created a number-based computer program that can formulate a phrase. Numbers with more than 500 digits were used to create the phrase; one included more than 3,000 digits.

Determining frequency often calls on a familiarity with big numbers. Treviño provided an example: determine the number of ways that numbers between 1 and 100 can be ordered.

Headlines are full of big numbers. The stock market's ups and downs are measured by economists, securities analysts and commodities agents. Research scientists and statisticians help determine the number of people affected by a surprise snowstorm. Physicists and astronomers identify and measure distances of the brightest comets and the closest planetary objects with water.