Representations of Numbers

Numbers exist around us ever day, whether they are used in Mathematics, Science, Accounting, or even Art. Often times numbers are represented in ways which make them more functional, such as binary numbers in computing, or easier to understand, as in a graph.

Numbers
Various basic representations of numbers. Tally marks, bar graph, and pie chart image sources referenced on final page.

In the above image there are depictions of various ways in which numbers can be viewed in our every day lives. On the left the number eleven is represented in three ways, our normal base 10 system, binary, and hexadecimal. The representations are used for function in Mathematics and computing, and are thus very simple. On the right are visual interpretations of numbers, such as tally marks for children to count, or graphs that can be used to compare different numbers. These types of examples are typically used in order to make numbers easier to understand in a general sense. The ways in which numbers are represented change based on the how the numbers are being used. As such, when manipulating numbers for a specific purpose one must adapt the representation to fit the audience and subject. Nancy Duarte depicts such an example in that, “A sales director presenting financial projections to a group of field reps wouldn’t visualize her data the same way that a design consultant would in a written proposal to a potential client.” It is my belief that by understanding how to properly represent numbers and data based on the context of a situation can lead to improved education in the areas of both Science and Art.

 

One might wonder, “What does Art have to do with numbers?”
Art is a form in which we represent numbers. In the same way that a child draws stick figures in a beginning Art class, children draw tally marks in their math classes. These tally marks are a form of Art. Similarly, bar graphs are a form of art that is used in comparing numbers visually, making it easy for students to determine which numbers are larger and which are smaller. In this way it is easy to see that Art and numbers are connected and can be used in driving education. Expanding this idea, how can Art be used to further education after a child’s early years? Evangelos Kapros presents the concept of using Art as a means of catching and holding on to the attention of students and motivating them to learn, “Motivating students is of paramount importance in a curriculum and cannot be overestimated. Visuals can be a stimulating experience for many students, and they should be used in a way that supports and is supported by learning design”.

 

By using Art to motivate and captivate students one can teach students Mathematics in a new way. Taking the concepts of binary numbers and visual representations of numbers and combining them, Ross Spencer has created a representation of prime numbers. This representation takes each prime number and converts it into a binary number. Each zero in the binary number is represented by a black spot in the image, while ones are represented by a colored tile. The colors are uniform within a number and randomly chosen between numbers.

primes-tiled
Spencer’s Tiled Prime Numbers
5-prime-numbers
1% of Tiled Prime Numbers (5 Primes)

 

This sort of visual representation of a complicated topic, prime numbers, can simplify the experience for students. In a similar fashion, this sort of technique could be used to introduce students to the topic of binary numbers, and would likely need to happen first. Students can find it difficult to comprehend binary numbers at first, and such a visualization could represent these numbers in a more familiar way. For example, binary addition could be simplified to comparing two colored tiles and creating a new tile following a set of rules. This visual representation of numbers and of operations can simplify complicated subjects and keep the attention of students more effectively than a series of ones and zeros.

 

While it is clear that numbers can be represented visually, and even as art, it is important to recognize that the same number may be represented in different ways. As the concept of data visualization becomes more widely known, both artists and data scientists will depict equivalent number in vastly different ways, similar to how two artists with their own unique styles will draw the same scene differently. A simple example of this is given below, in which the number Pi is depicted differently, despite being the same number and using the same visualization tool.

martin pi
The Art of Pi
vasile pi
Flow of Life Flow of Pi

In the first image Martin Kryzwinski represents Pi with both connections and circles, and in a more vibrant style than the following image. The second image is done by Christian Vasile. Both images were made using the same tool and represent the same number, yet without outside information you would believe these to be two separate numbers. While both of these images could be used in a classroom to depict the randomness and infinite nature of Pi, visual representations used in education could potentially confuse students. Students who see the first image in class and search for more information will likely see images such as the second or other images that look nothing like either of these, but still represent Pi. This could be a potential problem that may confuse students, but it could be handled similar to graphs and binary numbers. It is possible to create a pie graph and a bar graph that represent the same data just as you are able to represent 5 as 101 in binary. Still, the variability of data visualization is an issue that would have to be addressed in order to use this concept to aid in education.

As mentioned before, numbers are not the only thing that can be represented visually. Data can be turned into art, and has been for years. Below are examples of how another dimension can be added to number representation, wind maps are depicted in which the data consists of a number and direction.

windmap
Wind Map of US
earth windmap
Wind Map of Earth

 

This extra dimension is simply another parameter that is used to create the image. For the above examples the first dimension would be the wind speed, while the second is the wind direction. The direction would typically be interpreted as characters, such as NW or SSE, yet we can be more specific and represent these directions as numbers, more specifically angles. In this way we have wind maps that are derived from a set of 2 numbers, and is just an extension of the previous number visualizations.

 

It is my belief that by using these techniques of data visualization it is possible to improve education. Students can be more engaged in the classroom and more motivated to learn about topics that are taught using data visualization. Topics that are considered complicated, such as binary numbers, can also be simplified and thus taught more easily to a greater number of students by making use of these visualization techniques.

 

References

The Quick and Dirty on Data Visualization – Website

Data Visualization in and for Education – Website

Tally Marks – Website   Image

Bar Graph – Website   Image

Pie Chart – Website   Image

Tiled Prime Numbers – Website   Image 1   Image 2

The Art of Pi – Website   Image

Flow of Life Flow of Pi – Website   Image

Wind Map (US) – Website   Image

Earth Wind Map – Website   Image