3.4: Different Bases and Their Number Lines

The way we count, the way we visualize numbers, all starts from a number line. As early as Kindergarten, students learn how to read and use a number line. Our current number system is called the Decimal Number System (or Hindu-Arabic). Notice that Decimal begins with DEC like DECahedron (a 10 sided polygon) and we have 10 characters in our numbering system: zero through nine. So, even though we live in a Base 10 system (because we have characters for zero through nine), the number ten is represented by two characters (digits), one and zero.

Figure 2.5.1: Decahedron

Below is the number line for Base 10. Notice how it is broken up into rows. We will be using the number lines as rows for the sake of our lesson. However, when you teach the number line to the elementary school students, the number line will be one continuous row (to set up for negative numbers in second grade) OR set up like below only going from 1 to 10 or 0 through 10.

Figure 2.5.2: Number line for Base 10

There are an infinite amount of different bases and an infinite amount of corresponding number lines. Below are three different examples, written like we have Base 10 on the previous page.

Base 2 Number Line (reads from left to right, then top to bottom). The fifth number in the number line is \(101_{\text {two}}\).

Table 2.5.1: Base 2 Number Line

0	1
10	11
100	101
110	111
1000	1001
1010	1011
1100	1101
1110	1111
10000	10001

Base 8 Number Line (reads from left to right, then top to bottom). The \(10^{\text {th}}\) number is \(12_{\text {eight}}\).

Table 2.5.2: Base 8 Number Line

0	1	2	3	4	5	6	7
10	11	12	13	14	15	16	17
20	21	22	23	24	25	26	27

Base 12 Number Line (reads from left to right, then top to bottom). The \(15^{\text {th}}\) number is \(13_{\text {twelve}}\).

Table 2.5.3: Base 12 Number Line

0	1	2	3	4	5	6	7	8	9	A	B
10	11	12	13	14	15	16	17	18	19	1A	1B
20	21	22	23	24	25	26	27	28	29	2A	2B

Figure 2.5.3

Where are different bases used?

Base 2: Computers use Base 2, just zeros and ones for all of their programming. Your phone operates in only zeros (OFF) and ones (ON).

Base 5: This was one of the very first systems of counting, since we have five fingers.

Base 8: Think of Base 8 as the mathematics for the Cartoon Universe. Many Cartoons have only eight fingers. We have 10 fingers and live in a Base 10 universe. Cartoon have eight fingers and live in a Base 8 universe. The Cartoon character does not have a character for eight or nine items, as we do in our universe.

Base 12: Base 12 is rare to find within history. However, we currently have 12 hours on the clock and 12 months in the year. There were a few tribes in Africa and India which used the duodecimal (base 12) system.

Base 20: The Mayans used a Base 20 system, and invented the concept of zero.

Base 60: An extreme example is base 60, which was used by the Babylonians (about 4000 years ago) which is now current day Iraq. They had characters for 1 – 59 items. (The concept of “zero” was not discovered yet.) However, this is where our current concept of 60 minutes and 60 seconds come from.

Practice Problems

1. What is wrong about \(413_{\text {four}}\)?
2. How do you pronounce \(542_{\text {six}}\)?
3. Write four rows worth of the number line of Base 3.
4. If you were not familiar with the Base 10 system, which base system do you think would work best for our society and culture?