1. What Is a Number Base, and How Does Decimal Work?
Decimal is the system most people know. It uses ten digits, 0 through 9. When you reach 10, you move to the next place. For example, the number 365 means three 100s, six 10s, and five 1s: 3×100 + 6×10 + 5×1 [1].
A counting system where the place value increases every N steps is called "base N." The number N is called the "base." In base N, you use digits from 0 to N−1, and when you reach N, you move to the next place [1]. For instance, in binary (base 2), you only use 0 and 1. The binary number "10" equals the decimal number 2. So, what other bases do we use?
2. Why Do Clocks Use Base 60 and 12?
Minutes and seconds on a clock use base 60. This system traces back to the Babylonians in ancient Mesopotamia. They combined two types of wedge-shaped symbols to represent numbers from 1 to 59 [2].
Why choose 60? One theory is that 60 is divisible by 2, 3, and 5, making it easy to write fractions as finite numbers. Other theories suggest it came from mixing cultures that counted by 12 and 5, or from astronomy and geometry. There is no final answer yet [2]. Today, base 60 remains in time (60 seconds, 60 minutes) and angles (360 degrees in a circle) [2].
Base 12 is also common. The number 12 is divisible by 1, 2, 3, 4, 6, and 12, making it easy to divide into thirds or quarters. You see it in dozens (12), gross (144), clock faces (12 hours), months in a year, feet (12 inches), and the 12 animals of the Japanese zodiac [3]. Its origin is debated. Some say it comes from the roughly 12 full moons in a year, or its connection to Mesopotamian calendars [3]. These bases remained for various uses, partly due to their ease of division.
3. Why Do Computers Use Binary, or Base 2?
Computers use digital circuits that handle physical states like "high voltage" or "low voltage." They restrict themselves to just these 2 states, so binary (base 2) fits perfectly [4]. A single digit in binary is a "bit," and 8 bits make a "byte." For example, the decimal number 192 is 11000000 in binary [4].
The idea of binary is old. Around 200 BC, the Indian scholar Pingala described a way to count poetic rhythms using numbers, which is one of the earliest known descriptions of binary. The Chinese I Ching’s eight trigrams (3 bits) and 64 hexagrams (6 bits) can also be seen as binary sequences later [4]. In the 17th century, the mathematician Leibniz organized binary notation using 0 and 1, linking it to the I Ching trigrams. In 1937, Claude Shannon showed that switch circuits could be designed using Boolean algebra, forming the basis of computer theory [4].
Interestingly, the decimal number 0.1 cannot be written with a finite number of digits in binary. It becomes 0.0001100110011…, with the pattern repeating forever [4]. Binary numbers can also get long and hard for humans to read. How can we make them shorter?
4. How Does Hexadecimal Make Binary Easier to Read?
Long binary numbers are easier to read if you group them in fours. For example, 11000000 splits into 1100 and 0000. There are 16 possible combinations for four bits (2×2×2×2). If you assign one character to each combination, an 8-digit binary number shrinks to 2 characters. This is hexadecimal (base 16) [5].
Hexadecimal uses digits 0–9 and letters A–F (for 10–15). Since four binary digits equal one hex digit, it is shorter. One byte (8 bits) fits in 2 hex digits [5]. For example, 1100 is C (12) and 0000 is 0, so 11000000 is written as "C0" [5].
You see hexadecimal in web page colors. The code "#RRGGBB" sets the strength of Red, Green, and Blue. "FF" is 255 in decimal, the maximum value for one byte. "#FF0000" means maximum red, zero green, and zero blue, which is pure red [5]. Hex is also used to show computer memory contents. In programming, it might be written as 0x1000, depending on the language [5]. So, how do we convert between these bases?
5. How Do You Convert Numbers Between Decimal and Binary?
To change decimal to binary, divide by 2 and record the remainders. Then read the remainders backward. For 13: 13÷2=6 remainder 1, 6÷2=3 remainder 0, 3÷2=1 remainder 1, 1÷2=0 remainder 1. Reading backward gives 1101 [1].
To change binary to decimal, multiply each digit by its place value (1, 2, 4, 8…) and add them up. For 1101: 8 + 4 + 0 + 1 = 13 [1].
For binary and hexadecimal, just group binary digits in fours. 11000000 splits into 1100 (C) and 0000 (0), making C0 in hex. This equals 192 in decimal [5]. The decimal number 36864 is 9000 in hexadecimal [5]. All these systems use the same idea of place value. But where did place value and zero begin?
6. How Did the Maya Use Base 20 and a Symbol for Zero?
Different cultures used different bases. The ancient Maya used base 20. They used a dot for 1, a bar for 5, and a shell shape for 0. They wrote from bottom to top, with each place increasing by 20. For example, 33 is written as one dot (20) above two bars and three dots (13) [6].
For their calendar, they adjusted the second place to 17, and the third place to 360 (18×20) instead of 400 (20×20). This is explained as a way to get closer to the length of a year [6].
Where did place value and zero start? The Japanese Wikipedia states that place value notation came from ancient China. Decimal was used in the Shang Dynasty around the 14th century BC, and by the 4th century BC, zero was used to mark empty places [1]. The oldest document using decimal in Europe is a Spanish manuscript from 976 [1]. However, researchers debate whether the Maya invented zero first, or if the earlier Olmec civilization used it [6]. Different sources give different explanations, so this notebook cannot decide for sure.
7. How Can You Try Converting Numbers Yourself?
You can check these ideas with simple calculations. Try converting your age to binary. Divide by 2, record the remainders, and read them backward. You can check your answer by adding the powers of 2 to get back to your age.
Search for "#FF0000" to see that it is red. Try explaining why using the fact that "FF" is 255 [5]. Write 0.1 in binary by multiplying by 2 and taking the integer part repeatedly. You will see the repeating pattern [4]. If you want to learn more, write down one word that interests you, like "base" or "bit," as a starting point for further reading.