A Little History

Section 1.10 A Little History

Mathematics is a permanent fixture in human development. We find evidence of mathematical thinking in our most ancient of civilizations, and yet all modern technological advancements rely of good mathematics. Mathematics is the foundational tool we use to turn creative ideas into reality. But the human stories of mathematics is an area worth studying as well.

Throughout your coursework, you will learn about the interesting figures of math history and some of the incredible (and sometimes unbelievable) tales they are a part of. There are stories of young genius, incredible risks, underdogs, and duels. You will hear about mathematical giants from all over the world and how their contributions are still valuable to us today. Math history is rich and immensely interesting.

Subsection 1.10.1 What we take for granted

We learn about rationals, primes, and equations by the time we get to middle school, but so many of these ideas were unknown for generations. We take so much of our foundational mathematics for granted, but behind a lot of these concepts are moments of ingenious insight.

For example,

Our numerals are referred as "Hindu-Arabic numerals" invented by Indian mathematicians between the first and fourth centuries.
The discovery of irrational numbers threw the Pythagoreans into an existential crisis in the 4th century.
In the 17th century, mathematicians and philosophers Rene Descarte and Piere de Fermat independently connected geometry and algebra with a coordinate system we use in basic algebra today. This system is called "Cartesian coordiantes" after Descartes.

Figure 1.10.1. Portrait of Rene Descartes traditionally attributed to Frans Hals

Subsection 1.10.2 Thinking Differently

Studying math history also helps us see how culture impacts the way we think about things... even objective ideas such as mathematics. We can see that ancient cultures sometimes approached their problems from a perspective we might find unusual.

Subsubsection 1.10.2.1 Egyptian Multiplication

Figure 1.10.2. One of two surviving Egyptian mathematical papyri: The Rhind Papyrus

Ancient Egyptians were excellent at calculations, with applications in areas of economics, land area, and volumes of solids. They developed algorithms to help perform complicated calculations with high accuracy. We will look at their alogrithm for multiplying two integers.

For example, How would the Egyptians multiply 23 and 12? Their work all depends upon powers of two.

Question 1.10.3. Writing out Powers of 2.

Fill out the following tables with powers of two.

Table 1.10.4. Powers of 2

\(2^x\)	value
\(2^0\)
\(2^1\)
\(2^2\)
\(2^3\)
\(2^4\)

Solution.

Table 1.10.5.

\(2^x\)	value
\(2^0\)	1
\(2^1\)	2
\(2^2\)	4
\(2^3\)	8
\(2^4\)	16

The Egyption multiplication algorithm is sometimes referred to as the "double method", because the algorithm involves doubling one of the factors repeatedly. Let’s do this with the first factor, 23.

Question 1.10.6. Double 23.

Fill out the third column of the table by repeatedly doubling 23.

Table 1.10.7.

	\(2^x\)	\(23(2^x)\)
\(2^0\)	1	23
\(2^1\)	2
\(2^2\)	4
\(2^3\)	8
\(2^4\)	16

Solution.

Table 1.10.8.

	\(2^x\)	\(23(2^x)\)
\(2^0\)	1	23
\(2^1\)	2	46
\(2^2\)	4	92
\(2^3\)	8	184
\(2^4\)	16	368

Now we need to write our other factor as a sum of powers of two. For example, \(7 = 1 + 2 + 4 = 2^0 + 2^1 + 2^2\text{.}\)

Question 1.10.9. Powers of 2 to get 12.

Write 12 as a sum of powers of 2.

Solution.

\(12 = 4 + 8 = 2^2 + 2^3 \)

We determined earlier that \(12 = 2^2 + 2^3\text{,}\) and we just need to add the corresponding rows from Table 1.10.10.

Table 1.10.10.

	\(2^x\)	\(23(2^x)\)
\(2^0\)	1	23
\(2^1\)	2	46
\(2^2\)	4	92	\(\longleftarrow \)
\(2^3\)	8	184	\(\longleftarrow\)
\(2^4\)	16	368

Question 1.10.11. Our Final Calculation.

Add up the appropriate multiples of \(23\) to calculate \(12 \times 23\text{.}\)

Solution.

FromTable 1.10.10, we see the entries in rows corredsponding to \(2^2\) and \(2^3\) are \(92\) and \(184\text{.}\) So we conclude that \(12 \times 23 = 92 + 184 = 276\text{.}\)

Question 1.10.12. Another Multiplication Question.

Multiply \(15 \times 11 \) as would the Egyptians.

Subsubsection 1.10.2.2 Mathematics of Babylon

Unfortunately, due to the again of papyrus, we have few resources to pull from regarding Egyptian mathematics. However, the Babylonians baked their writings in cuneiform on clay tablets. This more durable writing system has provided us with a lot of information regarding Babylonian mathematics dating back to 1700 BC.

Figure 1.10.13. The Babylonian cuneiform tablet: Plimpton 322

Babylon made a lot of advancement in mathematics with knowledge of cubic equations and Pythagoras’ namesake, the Pythagorean Theorem (about 1000 years before Pythagoras), but we will just focus on the Babylonian number system.

Babylonians tracked numbers in base-60 (sexigesimal). Though that may sound odd, we still use sexigesimal systems when thinking about time (60 seconds to a minute) and angle measure (\(360^{\circ}\) in a circle).

So what does sexigesimal notation look like? Think about typing \(100\) on a microwave. After a second, what does that microwave read? Not \(99\text{,}\) right? The microwave knows that \(1\) minute means \(60\) seconds, and so after a second, we see \(59\text{.}\)

Similarly, the Babylonian would write \(1,00 \) for what we mean by \(60\text{.}\) We would read a number like \(11,32,54\) base-60 as \(11 \cdot 60^2 + 32 \cdot 60^1 + 54 \cdot 60^0 = 41,574\) base-10.

Question 1.10.14. Sexigesimal Practice.

Convert \(23,10,11\) into base-10.

Solution.

\(23,10,11 = 23\cdot 60^2 + 10 \cdot 60^1 + 11 \cdot 60^0 = 83,411.\)

Sexigesimal notation is particularly useful when dealing with fractions. For example, \(1;30\) is \(1\cdot 60^0 + 30 \cdot 60^{-1} = 1 + \frac{1}{2}\text{,}\) and by \(0;30;24\text{,}\) the Babylonians meant \(\frac{30}{60^1} + \frac{24}{60^2}\text{.}\)

To calculate \(\frac{a}{b} \text{,}\) Babylonians would first find \(\frac{1}{b}\) and then multiply it by \(a \text{.}\) For example, \(\frac{7}{2} \) is found first by finding \(\frac{1}{2} = 0;30 \) and then multiply the result by \(7 \) to get

\begin{equation*} \dfrac{7}{2} = 7 \cdot \dfrac{1}{2} = 7 \cdot (0;30) = 0;210 = 3;30. \end{equation*}

We finish our calculation by recognizing \(210 = 60\cdot 3 + 30 \text{.}\)

Question 1.10.15. Practicing with Sexigesimal Fractions.

Represent \(\dfrac{11}{4}\) as a sexigesimal fraction.

Subsection 1.10.3 Mathigon’s Timeline of Mathematics

Mathigon

mathigon.org/

has currated an interactive map of significant mathematical moments. We will now reference their timeline to answer the following questions.

Visit Mathigon’s Timeline of Mathematics

mathigon.org/timeline

Question 1.10.17.

Find a mathematician from Asia. What contribution did he or she make?

Question 1.10.18.

Find a mathematician listed between 1600 and 1800 Asia. What contribution did he or she make?

Question 1.10.19.

Find a mathematician listed between te 20th and 21st. What contribution did he or she make?

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