Early humans used tally marks and notches on bones and sticks to record quantities. The Ishango bone, found in Africa and dating back about 20,000 years, is one of the oldest known mathematical artifacts, showing that early humans had a sense of counting and number representation long before written language existed. This was the beginning of human mathematics!

The ancient Sumerians used a base-60 number system, known as the sexagesimal system. This is why we have 60 seconds in a minute and 60 minutes in an hour today. They also divided the circle into 360 degrees (6 × 60). Their system was so useful that we still use it for time and angles over 4,000 years later!

The Egyptians used a decimal (base-10) system with distinct hieroglyphic symbols for powers of 10: a vertical stroke for 1, a heel bone for 10, a coil of rope for 100, a lotus flower for 1,000, a finger for 10,000, a tadpole for 100,000, and a seated god for 1,000,000. They simply repeated symbols to add values. This system is similar to how Roman numerals work!

Ancient Indian mathematicians were the first to use zero as a digit in a number system, around the 5th century CE. The mathematician Brahmagupta wrote rules for using zero in calculations in 628 CE. Before this, cultures had placeholders but didn't treat zero as a number in its own right. The Indian number system with zero was later transmitted to the Islamic world and then to Europe, completely transforming mathematics.

The Babylonian system was a positional place value system, meaning the position of a symbol indicated its value (like our modern system). Using base-60, they could represent large numbers with fewer symbols than additive systems. However, they had no symbol for zero until later, which sometimes caused confusion. The earliest known use of a placeholder zero appears in Babylonian astronomical tablets from around 300 BCE.

Euclid (c. 300 BCE) is known as the 'Father of Geometry' for his work 'Elements,' a 13-volume treatise that organized all known geometry into a logical system of definitions, postulates, and proofs. It was the most influential mathematics textbook in history and was used for over 2,000 years. Euclid's 'Elements' was still being used as a textbook in schools in the 20th century!

The Mayan number system used base-20 (vigesimal) and was the first known civilization to use a symbol for zero in the Americas. They used a system of dots (representing 1), bars (representing 5), and shells (representing zero). Their base-20 system reflected counting on both fingers and toes. They used this number system to develop their famous calendar, astronomical calculations, and architectural measurements.

Roman numerals are an additive system, not a positional place value system. This means there is no concept of place value (like our tens, hundreds, thousands). Roman numerals also lack a symbol for zero, making it difficult to represent nothing as a value. Performing multiplication or division was extremely complicated using Roman numerals, which is why the adoption of the Indian-Arabic system was so revolutionary for European mathematics.

The Indian-Arabic number system is a positional place value system using base-10 and including zero as a digit. It allows numbers of any size to be represented with just ten symbols (0-9) and makes arithmetic simple and systematic. The system spread through the Islamic world via scholars like Al-Khwarizmi, whose works on algebra and Indian numerals became standard texts in Europe. The word 'algorithm' comes from his name! The system was so effective that it replaced all other number systems and is now the standard for mathematics worldwide.

Mathematical knowledge spread primarily through trade, conquest, and scholarship. The ancient Greeks traveled to Egypt and Mesopotamia to learn mathematics. Indian mathematics spread to the Islamic world through trade routes and scholarship. Islamic scholars translated Greek and Indian texts into Arabic, added their own discoveries, and transmitted this knowledge to Europe through Spain and other contacts. The Silk Road, the Islamic Golden Age, and the translation movement in places like Baghdad's House of Wisdom were all key to spreading mathematical ideas. Many mathematical works were preserved during the European 'Dark Ages' by Islamic scholars who translated and developed them further.