Babylonian Numbers

In an October blog “Decimals,” I showed you how we use the decimal (base-10) number system to generate the natural numbers we use today in math. In the blog “Mayan Numbers,” I showed you how the ancient Mayan civilization created numbers using a vigesimal (base-20) number system. In today’s blog, we’ll visit another ancient culture and learn how the Babylonian civilization used a sexegesimal (base-60) number system to do math.

Babylonia existed from about 18th c. BC until the 6th c. BC. This nation existed in the southern region of Mesopotamia between the Tigris and Euphrates River. Today, this area of the world is now part of Iraq.

The Babylonians used one of the earliest forms of written expression: the cuneiform script. They wrote in soft clay tablets using a wedge-shaped reed stylus. When the clay was hardened in the sun, the writing permanently became a part of the tablet.
They used just two symbols to define the 59 digits as follows:

The Babylonians wrote their numbers from right to left. The rightmost slot in any number has a multiple of 1. The multiplier for the next slots to the left of this column increased by a multiple of 60. The Babylonians defined their numbers by using one or more of the 59 digits from above. Although the Babylonians understood the concept of nothingness, they did not have a symbol for the digit or value of zero. They would leave a space or use a non-numeric placeholder to indicate that no digit was defined for a position within a number.

Now let’s use this knowledge to translate a Babylonian number (A).

We first translate the Babylonian digits into a Hindu-Arabic number and we multiply it by the multiplier associated with its position (B). We add the values of each position and come up with the value for the Babylonian number (C).

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The Vocabulary of Subtraction

Near the beginning of this month, there was a blog entry (“Vocabulary of Addition”) in which I defined the math terms for an equation using the operation of addition. Today, I will show you the English math words associated with the fundamental skill of subtraction and its associated numerical expressions {language model}.

A minuend is the number from which another is to be subtracted.

The minus sign is the symbol that represents the operation of subtraction in this equation.

The subtrahend is the number to be subtracted from the minuend.

The difference is the result of subtracting the subtrahend from the minuend.

As you can see, the terms associated with subtraction are a little more obscure and less commonly used in math conversations than the ones for addition. These terms emphasize the fact that the order of terms in subtracting two numbers does matter.  We need to recognize this difference when we use language to describe mathematics.

The Equality of Robert Recorde

Throughout their education, students will come across the most ubiquitous symbol in all of mathematics. It’s the symbol that defines all equations. It is the equal sign.  But where did it come from?

Robert Recorde (ca. 1512-1558) was a Welsh physician and mathematician. At the age of fifteen, he entered Oxford University, graduating in 1531. In the same year he was elected a Fellow of All Souls College. Moving to Cambridge University he obtained a degree in medicine in 1545. He is known as the founder of the English School of Mathematics and was the first person to write mathematical books in English. He also introduced, among other things, the idea of placing two hyphens in parallel to symbolize the balance of two expressions in a mathematical equation.

He wrote a math textbook with a most convoluted title: “The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers” (Published in 1557). You can find and download a digitized version of this textbook here at the Internet Archive. The book covers topics including whole numbers, the extraction of roots and irrational numbers as well as the execution of arithmetic operations.

This is the page in which he defines the familiar symbol:

“And to avoide the tedious repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one length, thus: =, because noe. 2. thynges, can be moare equalle.”

He uses this symbol to subsequently express numerous equations, beginning at the bottom of this page. Thus, it can be said that the first math equation to be expressed using this modern notation is 14x + 15 = 71. Thankfully, by the time this symbol was consistently used in the 1700s, the two parallel lines (as well as other symbols he defined) were made more compact.

Robert Recorde is honored with a wall tablet at St. Mary’s Church. It is located in his birth town of Tenby, Pembrokeshire, Wales. It documents his achievements to mathematics:

“To his genius we owe the earliest important English treatises on algebra, arithmetic, astronomy, and geometry: He also invented the sign of equality, =, now universally adopted by the civilized world.”

There is also a display at the Tenby Museum & Art Gallery documenting the achievements of this Welsh mathematical pioneer.

Explore and enjoy!

The Standard Algorithm of Addition: Part 3

In the blog “Standard Algorithm of Addition: Part 1,” we learned how to set up addition problems in a vertical format so that we can manually calculate their sum. We also learned the mathematical terms for the components of arithmetic expressions of additions. In the blog “Standard Algorithm of Addition: Part 2,” I used a {visual model} to explain the concept behind this fundamental algorithm using green square blocks and blue rods to represent the addends. In this blog, I will show you how the standard algorithm works {number model}.

First we set up the arithmetic expression in a vertical format (A). Now we will first focus on the ones column and work our way from right to left (B). We mentally add the digits in the ones (9+7=16). Note the sum is greater than 9. This sum is equivalent to 1 unit of ten (blue rod) and 6 units of ones (green squares) (C). I write the digit for the 6 units of one in the ones column under the underline (D). I place the 1 digit of ten at the top of the tens column (E).

Working our way from right to left, we move to the tens column (F). We add the digits in this column noting that we are actually adding multiples of ten (G). Since the sum is not greater than 99, we simply write the sum of the digits in the tens column underneath the underline (H). Therefore, the sum of the addends, 29 and 47, is 76 (I).  It is the most common technique of addition taught to math students, but it is not the only one.  We’ll look at some other addition algorithms in future blogs.

The Geometry of Flatland

Here is another literary reference to mathematics. In fact, this story has often been called mathematical fiction. It is the satirical novella “Flatland: A Romance of Many Dimensions,” (1884) which was written by Edwin Abbott Abbott (1838-1926), an English schoolmaster and theologian. This story explores the nature of dimensions from the point of view of a two-dimensional world of geometry. The narrator is the humble square, who ends up visiting the one-dimensional world of Lineland as well as the three-dimensional world of Spaceland.

He is even introduced to the world of Pointland. There are certainly some thought-provoking social elements to this geometry-filled story, and I recommend it to any math student who is studying geometry. Although the book was not ignored when it was published, it did not achieve great success. However, it experienced a surge in popularity when Albert Einstein’s Theory of General Relativity, and its introduction of the fourth dimension, was made known to the public. The text of the original books is made available to the public at WikiSource.
This story also inspired the making of a 30-minute animated movie, “Flatland: The Movie”. It was released and sold to the public as an educational edition DVD, private home-use DVD, and digital download in June, 2007. The educational edition is primarily intended for educators, teachers, schools and institutions that will use the movie as part of classes, lectures, and courses. Visit the movie’s website and watch the trailer. I found it to be a very contemporary and entertaining adaptation of this mathematical literary classic.  I hope you do too.

Standard Algorithm of Addition: Part 2

I believe that math students of all ages learn abstract notions by first experiencing them concretely with shapes they can actually see, touch and manipulate {visual model}. we will use of concrete teaching aids such as base-ten blocks help provide insight into the creation of the standard algorithm for addition. In this blog, we will use them to find the sum of two whole numbers, 29 and 47 (A). I will be using two types of blocks (B). A green square block has a value of 1 unit, and a blue rod has a value of 10 units. I represent the values of the two addends using these two blocks. I make sure that they are grouped in the appropriate columns according to the structure of the decimal number system (See “Decimals”). I then add all the blue rods together in the tens column, and add all the green squares together in the ones column (C). Notice that we have more than ten green squares. I group ten of these squares together (D) and replace them with a blue rod (E). This blue rod gets moved from the ones column to the tens column (F). Now we have seven blue rods and six green squares, which represent the sum: 76 (G).

We can use these two blocks to visually add numbers 1-99. Additional blocks can be added that represent larger multiples of 10 so that larger whole numbers can be used.

Can Alice do Addition?

John Tenniel (1820-1914) was a British illustrator, graphic humorist and political cartoonist. He achieved considerable fame as the illustrator of Alice. Tenniel drew ninety-two drawings for Lewis Carroll’s “Alice’s Adventures in Wonderland” (London: Macmillan, 1865) and “Through the Looking Glass” (London: Macmillan, 1871). There is an illustration to the ninth chapter of Through the Looking Glass and excerpt that touches upon the topic of recent blogs.

Illustration to the ninth chapter of “Through the Looking Glass” by John Tenniel. Wood-engraving by the Dalziels.

“Manners are not taught in lessons,” said Alice. “Lessons teach you to do sums, and things of that sort.”
“Can you do Addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?”

“I don’t know,” said Alice. “I lost count.”
“She can’t do Addition,” the Red Queen interrupted..

The .pdf copy of this book was made available to the public by Lenny de Rooy at her website, Lenny’s Alice in Wonderland Site.

Standard Algorithm of Addition: Part I

As part of our primary education in mathematics, we have all been taught a standard method for adding two or more numbers together. Students learn how to add a series of whole numbers by following a step-by-step process until they can apply it without making an error. As part of the math education, it is also important that math students understand why this step-by-step process, the standard algorithm of addition, works.

First we need to set up our arithmetic expression of addition properly {number model}. In the blog, “The Vocabulary of Addition,” we learned the math terms for the different components of arithmetic expressions and equations involving addition. In that blog, the arithmetic expression and equation was presented in a horizontal format. Such expressions and equations can also be presented in a vertical format. This is the format that is used to teach and implement the standard algorithm of addition.

The arithmetic expression, 89 + 47, is expressed in a vertical format.

Note how the digits in the one’s column and ten’s column are aligned vertically in this format.

The numbers listed in this format are the addends of the arithmetic expression of addition.

The plus sign, the symbol representing the operation of addition, is placed to the left of the addend at the bottom of the list.

The underline is used to separate the addends from the sum that is about to be calculated. It serves the same function as an equal sign.

When listing the addends associated with the arithmetic expression, the digits associated with the 1s column must line up vertically. The digits associated with the 10s column must also line up vertically. (See the blog “Decimals” for more details). This will make it easier to execute the algorithm properly. All math students should be encouraged to set up these addition problems properly and write them neatly to avoid confusion.

The most common way to gain insights into the validity of algorithms is to express its abstract concepts to the math student in visual concrete ways {visual model}. We’ll explore this concrete model of addition in Part II.

The Mechanical Arithmetic of the Pascaline

Blaise Pascal (1623-1662) is known today as a brilliant French mathematician, physicist, inventor, writer and Christian philosopher. In 1642, he attempted to help his father, a tax collector, deal with the repetitive arithmetic calculations that were part of the task of reorganizing the tax revenues of the French province of Upper Normandy. He was thus motivated to develop and invent the only functional calculator of the 17th century, known as the Pascaline, in 1645. It could add and (indirectly) subtract two numbers. It could also multiply and divide by repetition. In 1649 a royal privilege, by Louis XIV of France, gave him the exclusivity of the design and manufacturing of calculating machines in France. He designed the only functional calculator of the 17th century.

As far as I can tell, nine Pascalines still exist today. Four of them are on display at the National Conservatory of Arts and Crafts (CNAM) museum in Paris, France. There is also an interesting video that explains how this fascinating mechanical calculator worked. I encourage you to watch, learn and explore this intriguing bit of math history and technology.

The Vocabulary of Addition

In order to use language to communicate math concepts and skills concisely {language model}, we must learn the terminology and vocabulary of mathematics so that we can express ourselves accurately. Lets begin by learning the English math words associated with the fundamental skill of addition and its associated numerical expressions {number model}.

An arithmetic expression is a mathematical statement consisting of real numbers and mathematical operations. There are no variables in an arithmetic expression.

The numbers that are being added together are called addends.

The plus sign is the mathematical symbol representing the operation of addition.

The equal sign tells us that the the math statement on its left is the same value (equal) to the math statement on the right

The sum is the value of the addition of the addends.

This arithmetic equation is a mathematical statement which uses the equal sign to state that the arithmetic expression on the left (the sum of the two addends) is equal to the statement on the right of the equal sign (the sum).

Each language and culture uses its own terminology and vocabulary to represent these components of addition. Learning these words and their meaning is one important way the world of language arts and world of mathematics intersects.  One of the most basic tools I use to assess whether a student I am tutoring is fluent in the use of math vocabulary is through the use of flashcards.  I usually just ask the student to identify the component of the expression or equation highlighted in red.  Many times, there is more than one acceptable answer, as you will see in future blogs.