

We use base 10 because we have 10 fingers. In base 10, ten digits are used and those digits are 0 through 9. The Mayans used a vigesimal (base 20) number system, the Babylonians used a sexagesimal (base 60) number system, and the Egyptians used a duodecimal (base 12) number system. It is believed that the Mayans used base 20 because they lived in a warm climate where they did not wear shoes, thus giving 20 fingers and toes. The number 60 was significant to the Babylonians. The Babylonians came up with the concept of dividing a day into 24 hours with each hour having 60 minutes and each minute 60 seconds. The Babylonians also came up with the concept of a degree being divided into 60 minutes with each minute being divided into 60 seconds. Rather than counting fingers, the Egyptians counted the joints on each of the four fingers on a hand and there are three joints per finger giving twelve joints on a hand. 
The Mayas prospered in an area ranging from southern Mexico and the Yucatán Peninsula through Belize, Guatemala, Honduras, and El Salvador (Bazin, 2002). The first sign of their being in this area was around 2600 BCE. The Mayas built roads to connect their cities and built structures that are still considered amazing today (Bazin, 2002). They studied the cycles of the moon, earth, and other planets (Bazin, 2002). They built large temples covered in stucco, many still standing today. By the third century, one city, Palenque had its own modern drainage system and observatories (Bazin, 2002). It also had buildings that towered 110 feet above the jungle floor (Bazin 2002). The Spanish invaded the Mayan cities in the 1500s. During this time the Mayan books and writings were all destroyed (Bazin, 2002). Very little of their culture survived in written form due to these events. Scholars studied the minimal amounts of stone tablets with Mayan glyphs and deciphered that the Mayans used a number system of base 20 (Mayan Culture, 2010).
The Mayan numerals are shown to the left. The Mayan numbers are similar to our current way of keeping score with 4 lines then a slash through them to represent 5. They used a dot to represent one. The number three was represented by 3 consecutive dots. The number five was a straight bar. Six was a bar with one dot above it. There would be no more than 4 bars stacked (20).
The Mayans wrote in a stacked form. The chart to the left shows how the numbers 33, 429, and 5125 would be represented.
What is most intriguing about this culture is that they used a different base for the calendar year. When doing time calculations the least significant digit is multiplied by 20^{0}, but instead of the next digit to left being multiplied by 20^{1}, it is actually multiplied by 360. The next digit to the left is multiplied by 360 * 20 = 7200 and so on. Our numbers will be multiplied by 20^{n}x360 starting with the digit to the left of the least significant digit in our number. Imagine the challenge of writing a date as a number in a normal base 20 system. Can you write a date in both bases?
Video: Ancient Civilizations and Number Systems 
Babylonia was an ancient cultural region in centralsouthern Mesopotamia (presentday Iraq), with Babylon as its capital (Babylonia, 2010). The earliest mention of the city of Babylon can be found in a tablet dating back to the 23rd century BCE (Babylonia, 2010). The Babylonians made many clay structures due to the abundance of clay and lack of stone in the area (Babylonia, 2010). Their temples were massive structures made of brick with an actual drainage system during times of rain (Babylonia, 2010). They delved in astronomy as well but even more amazing is that they wrote books on medicine and even introduced the concepts of diagnosis, prognosis, prescription, and physical examination (Babylonia, 2010). Women and men both learned to read and write and each town had extensive libraries (Babylonia, 2010). They also showed evidence of the Pythagorean Theorem long before Pythagoras (Babylonia, 2010). Babylonian mathematics used a sexagesimal system (base 60). They used an array of symbols for numbers from 159 and there was no symbol for 0. These symbols are shown to the left.
The absolutely most fascinating thing that one can say about the Babylonian mathematics is they were able to compute the square root of 2 to 7 places (Babylonia, 2010), an accomplishment not truly computed till centuries later. The actual calculation was inscribed on a tablet that dates back to 1600 BCE (Babylonia, 2010) and duplicate tablets are shown to the left.
So how does base 60 work? Here are some examples of numbers from left to right.
Notice how this number may be written using numbers you know today. The number 4000 has 1 in the 3600s place, 6 in the 60s place, and 40 in the 1s place. Let us assume 40 can be represented by a value A. So numerically in Babylonia they would write 4000 as 16A. In what we use today that looks no more than ~160.
Egypt is the only civilization of the three that technically still exists today. The Mayan buildings are still standing, but the civilization is long gone with only descendants carrying on some of the traditions of the past. The Babylonian civilization has since fallen. Egypt is still a thriving country in the world today. The Egyptians used base 10 just as we do today (Bazin, 2002). The Egyptian numerals are shown to the left. The Egyptians were very skilled at math and even had interesting methods of doing things such as multiplication. They used powers of 2 to find out what the multiple of two numbers was (Bazin, 2002). They did a similar trick for division.