# Binary Counting System

One page for each of the four topics. Write a narrative (summary) of the historical development of each selected topic. Include contributions of significant figures and diverse cultures and how the topic originated and evolved over time.
1. Binary Counting System
2. Mathematic Differentiation

Acknowledge sources, using in-text citations and references in APA style, for content that is quoted, paraphrased, or summarized.

Example:
The historical development of the quadratic formula spans thousands of years. It is believed that between 3000 and 2000 BCE, Egyptian, Chinese, and Babylonian engineers developed quadratic equations in order to find the area of shapes. This was important to their societies for planning, building structures, and measuring fields. People in different parts of the world developed different solutions to these challenges over time. Babylonian mathematical tablets give us the first evidence of the solution to quadratic equations (Mastin, 2010). Measuring land was important in Babylonia as early as 3000 BCE to calculate taxes. The taxes owed were dependent on the number of crops that could be grown on a field, and quadratic equations were essential to determining the crop potential (Budd & Sangwin, 2004). Thus, the need to solve quadratic equations began due to the challenge of calculating taxes and building structures. The Egyptians estimated the areas for various shapes and sizes and noted them on papyrus (an old form of paper) around 1500 BCE. They then used the calculations as a lookup table to estimate the areas for shapes and sizes (Katz, 2004).

Over a thousand years later, around 400 BCE, the Babylonians used a more general method called “completing the square” to solve quadratic equations, using a numeric system, which made such calculations easier (Katz, 2004). Greek mathematician Pythagoras and Egyptian mathematician Euclid established more general procedures to solve the quadratic equation in 500 BCE and 300 BCE, respectively (Katz, 2004). An important factor in solving quadratic equations was the discovery of irrational numbers. Pythagoras discovered that the ratio of the area of a square to its side lengths is basically the square root and is not necessarily an integer. He dismissed this fact because the idea of a nonreal number was far too radical and ridiculous. In fact, it was a laughing matter. Pythagoras swore his close associates to secrecy to avoid hurting his reputation. One of the associates, however, released the concept of non-integers to the public. This angered Pythagoras, who had the associate executed (“An Essay on the Quadratic Formula,” n.d.). Then, in 300 BCE, Euclid found the proportion of a rectangle’s area to its side lengths may not result in an integer. He accepted this discovery, calling his solution irrational numbers (“An Essay on the Quadratic Formula,” n.d.).

Years later, in the seventh century CE, Indian mathematician Brahmagupta found the general solution for the quadratic equation, and the final complete solution was developed around 1100 CE by Indian mathematician Bhāskara (Fischbein & Brooks, 2006). In 1145 CE, Abraham bar Ḥiyya ha-Nasi (Savasorda) wrote a book that contained the complete solution to the quadratic equation. Finally, in 1637 CE, René Descartes published La Géométrie, which contained the quadratic formula that we know today (Katz, 2004). This history shows that humankind all over the world has a curiosity that has helped us solve practical problems and improve our way of life.

Katz, V. J. (2004). A history of mathematics. New York, NY: Pearson Addison Wesley.
Mastin, L. (2010). Sumarian/babylonian mathematics. The Story of Mathematics. Retrieved from www.storyofmathematics.com/sumerian.html

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