ishieldpoint: November 2022

Nov 8, 2022

X Squared: What Role Do Letters Play in Mathematics?

Mathematics is rooted in numbers, yet alphabetic symbols have long played a part in complex math and science. While it may not be immediately obvious how letters fit into mathematics, they are highly useful in the field. Letters help to simplify challenging problems, solve for unknown values, and make complex math accessible to everyday learners. What do letters represent in math? What types of letters are used? And how can you expect to see them applied in different equations?

What Do Letters Represent in Mathematics?

In mathematics, alphabetic letters function as placeholders for variables and represent functions. While we tend to think that letters only come into play within advanced mathematics, it might be useful to realize that letters can play a role in simple or basic math as well, where values are smaller and non-complicated. For example, you could ask someone what two plus two is. Or you could write out 2 + 2 = x, in which case that someone should determine that x = 4. A letter often represents a value that hasn’t been solved for yet, but not necessarily one that requires advanced techniques to solve.

There are many benefits to using letters in math. They allow mathematicians to simplify lengthy or convoluted equations to digestible formulas. This helps both students/educators and fellow mathematicians who might utilize or solve their formulas. Letters are also crucial for solving equations with missing or unidentified values and representing numbers that would not otherwise fit within the bounds of an equation (such as pi). Letters representing variables or functions are included on most advanced calculators. This helps users utilize their available tech to solve mathematical problems more efficiently.

What Types of Letters are Used in Math?

Mathematics largely relies on Roman letters and sometimes Greek letters. Using letters from a mixture of alphabets also allows for a wider representation of various numbers, functions, and variables. Mathematics commonly pulls from Roman and Greek lettering, though it also relies limitedly on other alphabets, such as Hebrew. Roman letters are found mainly in algebra, while Greek letters are common in other advanced math and science fields.

Roman Letters and Their Applications
As mentioned above, sometimes letters act as placeholders for values that are unknown or not yet identified. The letter x often serves this purpose, but w, y, and z are also commonly used in this way. Some Roman letters represent unknown or unsolved for values in very specific contexts. For example, the letter a is used to denote area. To solve for the area of a rectangle, one multiplies the length by the width of the shape. This formula is represented as a = l x w. In this case, a, l, and w all indicate unknown values, but also provide important context for solving the problem.

As formulas become more complex, more numbers become useful. If we’re solving for the dimensions of a 3d shape, for example, we might use V for volume. Volume would be the length times the width times the height of the shape, or put differently, V = l x w x h. Once you understand what the letters in a formula mean, their presence becomes helpful rather than confusing.

Greek Letters and Their Applications
Mathematics and related fields use all 24 Greek letters in equations, functions, and formulas. To non-Greek speakers, the application of these letters may be less intuitive, but learning them as letters rather than just symbols can be helpful for remembering their purpose. For example, ∑ often means sum. You could memorize that that shape means sum, or you could learn that letter as the Greek letter Sigma (the equivalent of a Roman alphabet’s “s”).

You may recognize this letter as the symbol for “sum” on a spreadsheet. For example, if you needed to add all of the numbers in the first 20 cells of a spreadsheet, you might click the sum button and be prompted to fill out something that looks like: ∑(1,20). This would tell your spreadsheet to add together all 20 values. This is how the sigma symbol is used in mathematics as well. Or rather, this is one real-life example of a mathematical equation in action.

Greek letters are highly functional in calculus, statistics, trigonometry, and the physical sciences. Most learners are first introduced to the Greek letter pi, which looks like π and is pronounced “pie”. Pi is the resulting value when you divide a circle’s circumference by its diameter. Pi is considered an infinite decimal, which makes it impossible to use just numbers to represent it in an equation (though in most cases, simply using 3.14 will get you close enough to the solution you need). Fun fact, last year Swiss researchers set a new world record by solving pi to its 62.8 trillionth decimal place.

In calculus, Δ, the Greek letter delta, represents a small change in a value and can be paired up with other variables to show limited changes in their size (Δx indicates a small change in x). Though all 24 Greek letters can be seen in mathematics, yet you are most likely to encounter these ones in common equations.

Hebrew letters are occasionally used in mathematics as well, though only at a very high level, so we won’t go into details here. Hopefully this explainer will help you decode the common letters likely to show up in the mathematics you may find yourself using in your everyday life.

First published on May 25, 2022, by David Naar
Culled from reference.com

Nov 4, 2022

The History of Independence Day in the U.S

By Nova Barela

Photo Courtesy: MediaNews Group / Orange County Register via Getty Images / Getty Images

Independence Day, or the Fourth of July, is celebrated in the United States to commemorate the Declaration of Independence and freedom from the British Empire. Although the Continental Congress voted in favor of independence from Great Britain on July 2, it wasn’t until July 4 that revisions of the declaration were completed. Most delegates didn’t sign the document until August 2.

Since then, Americans have celebrated the Fourth of July in many ways, although it wasn’t until 1870 that it was declared a holiday. From fireworks to parades, this is how — and why — Americans celebrate the Fourth of July

Early Celebrations

While people often celebrated on the Fourth of July in the decades after the Revolutionary War, there was little regularity to the festivities. John Adams described a spontaneous celebration in Philadelphia on the first anniversary of the original July Fourth, and both Philadelphia and Boston held the country’s first Fourth of July fireworks in 1777, but none of these festivities became annual celebrations. Bristol, Rhode Island held the oldest continuous Fourth of July celebration in 1785, but it wasn’t until after the War of 1821 that such events became common across the country.

Photo Courtesy: Spencer Platt / Getty Images

The first Fourth of July celebrations includes parades, speeches, feasts and especially toasting ceremonies. There had previously been a British tradition of using celebratory toasts on monarchs’ birthdays and other events to indirectly speak about current political events, and it was soon applied to the Fourth of July. By the mid-1790s, the Federalists and Democratic-Republicans — the two major political parties of the time — each held separate Fourth of July events where the festivities had distinctly political overtones. In the 1800s, the Fourth was used by everyone from abolitionists to nativists to spread their political goals.

The Fourth of July represented not just a time to celebrate the country’s past, but also to plan its future. While it wasn’t yet a federal holiday, it was an important celebration, with one European observer even going so far as to describe it as “almost the only holy-day kept in America.”

Black Americans and Independence Day

For much of U.S. history, Black American enthusiasm for Independence Day was mild at best, and for understandable reasons: the American Revolution brought freedom from the British crown, it did not end slavery. In fact, many of the Founding Fathers themselves owned slaves, while the British abolished slavery three decades before the United States in 1833.

Photo Courtesy: Library of Congress / Getty Images

In a speech later known as “What to the Slave Is the Fourth of July?,” abolitionist and former slave Frederick Douglass eloquently critiqued the holiday: “The rich inheritance of justice, liberty, prosperity and independence, bequeathed by your fathers, is shared by you, not by me. The sunlight that brought life and healing to you, has brought stripes and death to me. This Fourth [of] July is yours, not mine. You may rejoice, I must mourn.”

After the Civil War, Black Americans suddenly had cause to embrace the holiday, with massive celebrations sweeping the South. White Americans, however, resented their participation in the Fourth of July. Through a combination of new laws and acts of violence, Black celebrations of Fourth of July became a thing of the past by the end of the 19th century.

Fourth of July Fisticuffs

While we don’t think of the Fourth of July as a particularly violent holiday today, wrestling and other forms of controlled fights were once popular ways to celebrate American independence. One account in the Library of Congress from a woman who lived in Oregon during the 1870s describes a tradition where “the men who drank got into fights and called each other Englishmen” in a rather creative reenactment of the American Revolution.

Photo Courtesy: DEA / M. SEEMULLER / Getty Images

Similarly, a South Carolina man describes a sort of Fourth of July fighting tournament: “A great barbecue and picnic dinner would be served; candidates for military, state, and national offices would speak; hard liquor would flow; and each section would present its ‘bully of the woods’ in a contest for champion in a fist and skull fight. Butting, biting, eye gouging, kicking, and blows below the belt were barred. It was primitive prize fighting.”

Big boxing matches, such as the July 4, 1910 World Heavyweight Championship match between Jack Johnson and James Jeffris (illustrated above), were also major draws on Independence Day. Needless to say, Fourth of July Celebrations of years past could get a bit rowdy.

Independence Day Now

Today’s Fourth of July celebrations still emphasize the founding of the country, independence from Britain and the liberties Americans have won over time. However, the observance of the holiday has shifted over time. Deaths, injuries and property destruction caused by fireworks, alcohol consumption and even tetanus-inducing toy guns led to the Safe and Sane Fourth of July movement, which called for less dangerous Independence Day celebrations. As a result, the Fourth of July today is much tamer than it once was.

Photo Courtesy: Joshua Lott / Getty Images

Additionally, while speeches from elected officials were once an essential part of the holiday, few people spend Independence Day focused on politics. Instead, because of a rising standard of living and a shorter work week thanks to the labor movement, many Americans use the holiday as a chance to enjoy summer and hold cookouts with friends and family. Freedom and patriotism are still the reasons for the season, but hotdogs and hamburgers fresh off the grill also help keep Americans enthusiastic about Independence Day.