Systems of Numeration: Hindu-Arabic, Babylonian, Mayan, and Base Conversions

Systems of Numeration

Hindu-Arabic Numeration System

The Hindu-Arabic Numeration System is the most widely used system in the world today. It is a positional system, meaning the value of a digit depends on its position within the number. This system uses ten digits (0–9) and is based on powers of ten.

• Positional System: Each digit's value is determined by its place value (ones, tens, hundreds, etc.).

• Expanded Form: Numbers can be written as a sum of each digit multiplied by its place value. Example: $5,893 = (5 \times 1000) + (8 \times 100) + (9 \times 10) + (3 \times 1)$

• Conversion: To convert from expanded form to a Hindu-Arabic numeral, combine the products of each digit and its place value.

• Example: Convert $(4 \times 10^5) + (6 \times 10^3) + (7 \times 10^2)$ to a Hindu-Arabic numeral: $4 \times 100000 = 400000$ $6 \times 1000 = 6000$ $7 \times 100 = 700$ Total: $400000 + 6000 + 700 = 406700$

Babylonian Numeration System

The Babylonian Numeration System is an ancient system based on base 60 (sexagesimal). It uses unique symbols to represent values, and the place value increases by powers of 60.

• Base 60: Place values are powers of 60 (1, 60, 3600, ...).

• Symbols: The system uses specific symbols for 1 and 10, and combinations thereof for other values.

• Zero: Zero is represented by spaces or a special symbol.

• Conversion: To convert Babylonian numerals to Hindu-Arabic numerals, multiply each symbol's value by its place value and sum the results.

• Example: Convert a Babylonian numeral to Hindu-Arabic: $\text{Babylonian symbol for 2} \times 60^2 + \text{symbol for 5} \times 60^1 + \text{symbol for 1} \times 60^0$

Mayan Numeration System

The Mayan Numeration System is a vigesimal (base 20) system with unique symbols for 1, 5, and 0. Mayan numerals are written vertically, with each level representing a different place value.

• Place Values: The Mayan system uses place values of 1, 20, 400, 8000, etc., but the base is not always consistent.

• Symbols: Dots represent 1, bars represent 5, and a shell symbol represents 0.

• Vertical Notation: Numerals are written from top to bottom, each level corresponding to a place value.

• Conversion: Multiply each symbol's value by its place value and sum the results to get the Hindu-Arabic equivalent.

• Example: Represent 751 as a Mayan numeral: $751 = (1 \times 400) + (18 \times 20) + (11 \times 1)$

Converting Between Bases

Converting from Bases Below 10 to Base 10

To convert a number from a base less than 10 to base 10, use the expanded form and sum the products of each digit and its place value.

• Steps:

  1. Identify the base and place values.

  2. Write the number in expanded form.

  3. Multiply and add to get the base 10 equivalent.

• Example: Convert $1011_2$ to base 10: $1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11$

Converting from Bases Above 10 to Base 10

For bases above 10, digits greater than 9 are represented by letters (A=10, B=11, C=12, etc.). The conversion process is similar, using expanded form.

• Steps:

  1. Identify the base and place values.

  2. Write the number in expanded form, converting letters to their numeric values.

  3. Multiply and add to get the base 10 equivalent.

• Example: Convert $2AD_{16}$ to base 10: $2 \times 16^2 + 10 \times 16^1 + 13 \times 16^0 = 512 + 160 + 13 = 685$

Converting from Base 10 to Other Bases

To convert a base 10 number to another base, repeatedly divide by the new base and record the remainders.

• Steps:

  1. Identify the base to convert to.

  2. Divide the base 10 number by the new base, recording the quotient and remainder.

  3. Continue dividing the quotient by the base until the quotient is zero.

  4. The remainders, read in reverse order, form the new base number.

• Example: Convert 45 to base 2: $45 \div 2 = 22$ remainder 1 $22 \div 2 = 11$ remainder 0 $11 \div 2 = 5$ remainder 1 $5 \div 2 = 2$ remainder 1 $2 \div 2 = 1$ remainder 0 $1 \div 2 = 0$ remainder 1 Read remainders in reverse: 101101

Performing Computations in Different Bases

Addition in Other Bases

To add numbers in bases other than 10, add the digits as in base 10, but convert the sum to the indicated base and carry as needed.

• Carry: If the sum of digits is greater than or equal to the base, carry to the next place value.

• Check: Convert the result to base 10 to verify correctness.

• Example: Add $1011_2 + 1101_2$ in base 2.

Subtraction in Other Bases

Subtraction in other bases follows the same process as in base 10, but when borrowing, the number borrowed is the base.

• Borrow: If the digit being subtracted is larger, borrow the base from the next place value.

• Check: Convert to base 10 to verify correctness.

• Example: Subtract $1011_2 - 1101_2$ in base 2.

Multiplication in Other Bases

Multiply digits in base 10, then convert the product to the indicated base and carry as needed. For multi-digit numbers, use the same process as in base 10.

• Carry: If the product is greater than or equal to the base, carry to the next place value.

• Check: Convert to base 10 to verify correctness.

• Example: Multiply $101_2 \times 11_2$ in base 2.

Division in Other Bases

Division in other bases uses the same steps as long division in base 10, but all calculations are performed using the rules and digits of the indicated base.

• Remainder: When no more digits can be brought down, the remaining value is the remainder.

• Check: Multiply the quotient by the divisor and add the remainder to verify correctness in base 10.

• Example: Divide $1010_2 \div 10_2$ in base 2.

Summary Table: Numeration Systems Comparison

System	Base	Symbols	Place Values	Example
Hindu-Arabic	10	0–9	1, 10, 100, ...	5,893
Babylonian	60	Special symbols for 1, 10	1, 60, 3600, ...	147
Mayan	20 (not always consistent)	Dots, bars, shell	1, 20, 400, ...	751