• Understand different number systems
• Convert different number systems

Number Bases

Having explored the fascinating history of counting systems, we now turn our focus to an innovation that some would argue is even more impactful on mathematics: the development of place value systems and various bases. Let’s dig into how different number bases have evolved throughout history, starting with some of the earliest civilizations. While the base-[latex]10[/latex] system might be the reigning champ in the modern world (thanks to our ten fingers), various bases have been used throughout history.

Ever wondered why an hour has [latex]60[/latex] minutes or a circle has [latex]360[/latex] degrees? You can thank the ancient Babylonians for that. Although it is in slight dispute, the earliest known document in which the Indian system displays a positional system dates back to 346 CE. However, some evidence suggests that they may have actually developed a positional system as far back as the first century CE. Living in what’s now southern Iraq around [latex]4,000[/latex] years ago, the Babylonians used a base-[latex]60[/latex], or sexagesimal, number system. This system, which greatly contrasts our conventional base-[latex]10[/latex] system, was adopted due to [latex]60[/latex]‘s status as a “highly composite number.”  A highly composite number is characterized by having more divisors than any smaller positive integer.  In this case,  [latex]60[/latex] can be evenly divided by [latex]1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30,[/latex] and [latex]60[/latex]. However, there is not much evidence that the Babylonian system had much impact on later numeral systems, except with the Greeks.

The Babylonians were not the only civilization to use a different base number system from the traditional base-[latex]10[/latex] system. The ancient Mayan civilization held a preference for the base-[latex]20[/latex], or vigesimal, system. This choice may have been influenced by using fingers and toes for counting. Interestingly, their complex calendar system, which became infamous due to misinterpretations related to the year 2012, was predicated on a hybrid of base-[latex]18[/latex] and base-[latex]20[/latex] calculations.

The Chinese had a base-[latex]10[/latex] system, probably derived from the use of a counting board.^[1] Interestingly, it is suggested by some scholars that the positional system used in India was derived from this Chinese base-10 system. This offers a fascinating glimpse of the exchange and evolution of mathematical concepts along the ancient Silk Road.

The Ancient Greeks had a unique approach to mathematics and their numbering system reflected this. Instead of using a base system like many of the civilizations we’ve discussed, the Greeks used a system where each numeral represented a different number, similar to the Roman numeral system. This system, known as the Greek numeral system, is believed to have been developed around the 6th century BC. It involved the use of the Greek alphabet, and a few other characters, to denote numbers. In this system, the first nine letters of the Greek alphabet represented the numbers [latex]1[/latex] to [latex]9[/latex], the next nine represented the multiples of [latex]10[/latex] ([latex]10[/latex] to [latex]90[/latex]), and the last nine represented the hundreds ([latex]100[/latex] to [latex]900[/latex]).

It’s fascinating to see how the Greeks, just like the Sumerians, Mayans, Chinese, and others, developed a numbering system that suited their specific needs and contributed to their unique mathematical legacy.

A discussion of base systems would be incomplete without mentioning the base-[latex]2[/latex], or binary system. This system, comprised entirely of [latex]1[/latex]s and [latex]0[/latex]s, is foundational to modern computing technology. At first glance, binary might appear alien when compared to our everyday system, but it is ideally suited for electronic devices that operate by switching electricity on ([latex]1[/latex]) and off ([latex]0[/latex]).

Contemplating the origins of these diverse base systems highlights the pragmatic nature of their development, whether based on physical attributes such as fingers and toes or to simplify calculations. Despite the predominance of base-[latex]10[/latex] in current use, the world of mathematics is much richer and historically deeper than one might initially consider. Recognizing a [latex]60[/latex]-minute hour or a string of binary code is, therefore, an engagement with the history of human mathematical endeavor.

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