Algebra uses symbols come represent quantities without fixed values, known as variables.(Image credit: Volt repertoire | Shutterstock)
Algebra is a branch that mathematics handling symbols and also the rules for manipulating those symbols. In primary school algebra, those symbols (today composed as Latin and also Greek letters) represent amounts without solved values, known as variables. Simply as sentences describe relationships between specific words, in algebra, equations describe relationships between variables. Take the following example:

I have actually two fields that total 1,800 square yards. Returns for each field are ⅔ gallon that grain every square yard and also ½ gallon every square yard. The first field provided 500 more gallons than the second. What room the locations of every field?

It"s a popular notion that such troubles were invented to torment students, and also this might not be far from the truth. This difficulty was nearly certainly composed to help students understand mathematics — yet what"s special about it is it"s virtually 4,000 years old! follow to Jacques Sesiano in "An development to the background of Algebra" (AMS, 2009), this trouble is based on a Babylonian clay tablet circa 1800 B.C. (VAT 8389, Museum of the ancient Near East). Since these root in ancient Mesopotamia, algebra has actually been central to many breakthroughs in science, technology, and civilization together a whole. The language the algebra has actually varied significantly throughout the background of all human beings to inherit it (including our own). Now we write the trouble like this:

x + y = 1,800

The letters x and also y stand for the locations of the fields. The first equation is taken simply together "adding the two locations gives a complete area the 1,800 square yards." The second equation is much more subtle. Since x is the area of the very first field, and the first field had a yield of two-thirds the a gallon every square yard, "⅔∙x" — meaning "two-thirds times x" — represents the total amount the grain developed by the very first field. Similarly "½∙y" to represent the complete amount that grain created by the second field. Because the first field offered 500 much more gallons the grain 보다 the second, the distinction (hence, subtraction) between the very first field"s serial (⅔∙x) and also the 2nd field"s grain (½∙y) is (=) 500 gallons.

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Answer mr out

Of course, the strength of algebra isn"t in coding statements around the physics world. Computer system scientist and author mark Jason Dominus writeson his blog,The world of Discourse: "In the very first phase you interpret the trouble into algebra, and then in the 2nd phase girlfriend manipulate the symbols, nearly mechanically, till the answer mister out together if by magic." While this manipulation rules derive from mathematical principles, the novelty and also non-sequitur nature that "turning the crank" or "plugging and also chugging" has actually been i found it by countless students and professionals alike.

Here, we will settle this problem using approaches as they are taught today. And as a disclaimer, the leader does not need to recognize each certain step to grasp the prominence of this all at once technique. It is my intention the the historical significance and also the fact that we space able to solve the problem without any type of guesswork will inspire inexperienced reader to learn around these measures in higher detail. Right here is the an initial equation again:

x + y = 1,800

We solve this equation because that y by individually x fromeach side of the equation:

y = 1,800 – x

Now, we bring in the 2nd equation:

⅔∙x – ½∙y = 500

Since we found "1,800 – x" is same to y, it may besubstitutedinto the second equation:

⅔∙x – ½∙(1,800 – x) = 500

Next,distributethe an adverse one-half (–½) throughout the expression "1,800 – x":

⅔∙x + (–½∙1,800) + (–½∙–x) = 500


⅔∙x – 900 + ½∙x = 500

Add the two fractions that x together and add 900 toeach side of the equation:

(7/6)∙x = 1,400

Now, divideeach side of the equationby 7/6:

x = 1,200

Thus, the very first field has an area of 1,200 square yards. This value might besubstitutedinto the first equation to recognize y:

(1,200) + y = 1,800

Subtract 1,200 fromeach next of the equationto fix for y:

y = 600

Thus, the second field has actually an area of 600 square yards.

Notice how often we rental the an approach of doing an procedure toeach next of an equation. This practice is best understood together visualizing an equation together a scale with a well-known weight top top one side and an unknown load on the other. If we include or subtract the exact same amount of load from each side, the scale remains balanced. Similarly, the scale remains balanced if us multiply or division the weights equally.

While the an approach of maintaining equations balanced was practically certainly offered by all people to breakthrough algebra, using it to fix this old Babylonian trouble (as displayed above) is anachronistic due to the fact that this an approach has only been main to algebra because that the last 1,200 years.

prior to the middle Ages

Algebraic thinking underwent a an extensive reform following the breakthrough by scholar of Islam"s gold Age. Until this point, the civilizations that inherited Babylonian mathematics practiced algebra in progressively sophisticated "procedural methods." Sesiano more explains: A "student needed to memorize a small variety of identities, and also the art of resolving these troubles then consist of in transforming each problem into a standard kind and calculating the solution." (As one aside, scholars from old Greece and India did exercise symbolic language come learn about number theory.)

An Indian mathematician and astronomer, Aryabhata (A.D. 476-550), wrote among the earliest-known books on math and astronomy, dubbed the "Aryabhatiya" by modern-day scholars. (Aryabhata did no title his work-related himself.) The work-related is "a small astronomical writing written in 118 verses offering a summary of Hindu mathematics up to the time," follow to theUniversity the St. Andrews, Scotland.

Here is a sample of Aryabhata"s writing, in Sanskrit. This is city 2.24, "Quantities from their difference and product":


Aryabhatiya, city 2.24: "Quantities from their difference and also product." Sanskrit, palm leaf, A.D. 499. (Image credit: Robert Coolman)

According come Kripa Shankar Shukla in "Aryabhatiya the Aryabhata" (Indian national Science Academy of brand-new Delhi, 1976), this verse about translates to:

2.24: To identify two quantities from their difference and also product, multiply the product by four, then include the square of the difference and also take the square root. Write this an outcome down in two slots. Boost the an initial slot through the difference and also decrease the 2nd by the difference. Cut each slot in fifty percent to obtain the worths of the 2 quantities.

In modern-day algebraic notation, we compose the difference and product favor this:

x – y = A (difference)

x∙y = B (product)

The procedure is then written prefer this:

x = < √(4∙B + A2) + A >/2

y = < √(4∙B + A2) - A >/2

This is a sports of the quadratic formula. Comparable procedures appear as far earlier as Babylonia, and represented the state the algebra (and its nearby ties to astronomy) for much more than 3,500 years, throughout many civilizations: Assyrians, in the 10th century B.C.; Chaldeans, in the seventh century B.C.; Persians, in the 6th century B.C.; Greeks, in the 4th century B.C.; Romans, in the first century A.D.; and Indians, in the 5th century A.D.

While together procedures virtually certainly originated in geometry, it is necessary to keep in mind the initial texts native each people say for sure nothing around how such procedureswere determined, and also no initiatives were do toshowproofof your correctness. Written records addressing this problems an initial appeared in the center Ages.

Algebra"s adolescence

TheGolden period of Islam, a period from the mid-seventh century come the mid-13th century, experienced the spread of Greek and Indian mathematics to the Muslim world. In A.D. 820,Al-Khwārizmī, a faculty member the the home of Wisdom that Baghdad, released "Al-jabr wa"l muqabalah," or "The Compendious publication on calculation by Completion and Balancing." the is indigenous "al-jabr" that we derive our word "algebra." Al-Khwārizmī also developed quick methods because that multiplying and also dividing numbers, i m sorry are known as algorithms — a corruption that his name. He additionally suggested that a tiny circle need to be used in calculations if no number appeared in the tens ar — thusinventing the zero.

For the very first time due to the fact that its inception, the exercise of algebra change its emphasis away fromapplyingprocedural methods more toward means ofproving and also derivingsuch methods using geometry and also the method of act operations to every side of an equation. Follow to Carl B. Boyer in "A background of math 3rdEd." (2011, Wiley), Al-Khwārizmī uncovered it "necessary that we should show geometrically the fact of the same problems which us have explained in numbers."

Medieval Muslim scholars wrote equations out together sentences in a tradition now known asrhetoricalalgebra. Over the following 800 years, algebra advanced over a spectrum of rhetorical and also symbolic language well-known assyncopatedalgebra. The pan-Eurasian heritage of expertise that included mathematics, astronomy and also navigation uncovered its method to Europe in between the 11thand 13thcenturies, primarily through the Iberian Peninsula, i m sorry was recognized to the Arabs together Al-Andalus. Particular points of transmission to Europe to be the 1085 conquest of Toledo through Spanish Christians, the 1091 re-claiming of Sicily by the norman (after the Islamic occupation in 965) and the Crusader war in the Levant native 1096 to 1303. Additionally, a number of Christian scholars such as Constantine the african (1017-1087), Adelard of bath (1080-1152) andLeonardo Fibonacci(1170-1250) travel to Muslim soil to discover sciences.

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Fully symbolic algebra — together demonstrated in ~ the beginning of the short article — wouldn"t be recognizable until the scientific Revolution. René Descartes (1596-1650) used algebra us would recognize today in his 1637 publishing "La Géométrie," which pioneered the practice of graphing algebraic equations. According to Leonard Mlodinow in "Euclid’s Window" (Free Press, 2002), Descartes" "geometric approaches were so critical to his insights the he composed that "my entire physics is nothing other than geometry."" Algebra, having actually departed from its procedural geometric partner 800 years previously to develop into a symbolic language, had actually come complete circle.