Understanding the Mayan Numeration System: Principles and Conversions and Place Values

The Mayan numeration system is the final system discussed in the series, following the Egyptian, Babylonian, Roman, and Chinese systems.
This system is most closely related to the Babylonian numeration system because it utilizes a place value table.
While the Babylonian system used a base of $60$ , the Mayan system is described as "almost" a base $20$ system.
A key distinction of the Mayan system is its vertical orientation; place values are arranged from bottom to top rather than horizontally.

The Mayan system uses only three distinct symbols to represent all numbers: * Zero: Represented by a symbol that looks like a shell, a "loaf of bread", or a "football". * One: Represented by a single dot. * Five: Represented by a horizontal line.
Stacking Rules: Symbols are stacked within each place value tier. * Dots are always placed on top of lines within a specific place value. * For example, the number $14$ is represented as two horizontal lines (totaling $10$ ) with four dots stacked on top. * The number $19$ is represented as three horizontal lines (totaling $15$ ) with four dots stacked on top.
Place Value Separation: If dots are found underneath lines, it indicates the start of a different (lower) place value tier.

Mayan numerals are read from bottom to top, with each level representing a higher power or multiple.
The First Position (Bottom Tier): Represents the ones place, or $20^0$ . Any value in this tier is multiplied by $1$ .
The Second Position: Represents the $20$ s place, or $20^1$ . Any value in this tier is multiplied by $20$ .
The Third Position (The "Twist"): This position is not $20^2$ ( $400$ ) as would be expected in a pure base $20$ system. Instead, it is $20 \times 18$ , which equals $360$ . * Rationale: The Mayans were expert astrologers and mathematicians. They considered the number $360$ highly significant for their calendar and astronomical observations, so they adjusted their number system to include it.
Subsequent Positions: After the third position, the system continues to increase by factors of $20$ . * The fourth position is $20^2 \times 18$ , which equals $7,200$ . * The fifth position is $20^3 \times 18$ , which equals $144,000$ , and so on.

To convert from Mayan to Hindu-Arabic, first identify the values represent by the symbols in each vertical tier.
Multiply the symbol value in each tier by that tier's respective place value.
Sum all the resulting products together to find the final Hindu-Arabic number.

Symbols: A single dot in the top tier and the "loaf of bread" (zero) in the bottom tier.
Base Setup: * Top Tier: $20^1 = 20$ * Bottom Tier: $20^0 = 1$
Equation: * $(1 \times 20) + (0 \times 1) = 20$
Final Answer: $20$

Symbols: * Top Tier: One dot ( $1$ ) * Middle Tier: One line and one dot ( $6$ ) * Bottom Tier: One line and one dot ( $6$ )
Base Setup: * Top Tier (Position 3): $20 \times 18 = 360$ * Middle Tier (Position 2): $20^1 = 20$ * Bottom Tier (Position 1): $20^0 = 1$
Equation: * $(1 \times 360) + (6 \times 20) + (6 \times 1) = 360 + 120 + 6 = 486$
Final Answer: $486$

Symbols: * Position 5 (Top): One line and one dot ( $6$ ) * Position 4: The bread/football symbol ( $0$ ) * Position 3: Two lines ( $10$ ) * Position 2: Two lines and two dots ( $12$ ) * Position 1 (Bottom): Two lines and four dots ( $14$ )
Equation: * $(6 \times 20^3 \times 18) + (0 \times 20^2 \times 18) + (10 \times 20 \times 18) + (12 \times 20^1) + (14 \times 1)$
Calculated values: * $(6 \times 144,000) + 0 + (10 \times 360) + (12 \times 20) + 14$ * $864,000 + 0 + 3,600 + 240 + 14 = 867,854$

To convert from Hindu-Arabic to Mayan, divide the number by the base value (usually starting with $20$ for smaller numbers).
Perform long division to determine the quotient and the remainder.
The remainder represents the value for the lower tier (ones place).
The quotient represents the value for the next higher tier.

Step: Divide $28$ by $20$ . * $20$ goes into $28$ one time ( $1$ ) with a remainder of $8$ .
Symbol Application: * Bottom level (ones place): Represent the remainder ( $8$ ) using one line and three dots. * Top level ( $20$ s place): Represent the quotient ( $1$ ) using one dot.

Step: Divide $350$ by $20$ . * $20$ goes into $350$ seventeen times ( $17$ ). * $17 \times 20 = 340$ . * $350 - 340 = 10$ (remainder).
Symbol Application: * Bottom level (ones place): Represent the remainder ( $10$ ) using two lines. * Top level ( $20$ s place): Represent the quotient ( $17$ ) using three lines and two dots.

Step: Divide $256$ by $20$ . * $20$ goes into $25$ once ( $1$ ). * $20$ goes into $56$ twice ( $2$ ). * Total quotient is $12$ . * $12 \times 20 = 240$ . * $256 - 240 = 16$ (remainder).
Symbol Application: * Bottom level (ones place): Represent the remainder ( $16$ ) using three lines and one dot. * Top level ( $20$ s place): Represent the quotient ( $12$ ) using two lines and two dots.

1. Can we look at a question together on converting to Hindu-Arabic for the value 3,000?

Prompt/Question: Given a Mayan number with three tiers: * Top Tier (Position 3): One line and three dots ( $8$ ). * Middle Tier (Position 2): One line and one dot ( $6$ ). * Bottom Tier (Position 1): The bread symbol ( $0$ ).
Response/Process: * Set up the calculation for Position 3 ( $20 \times 18 = 360$ ), Position 2 ( $20$ ), and Position 1 ( $1$ ). * Equation: $(8 \times 360) + (6 \times 20) + (0 \times 1)$ . * Calculation: $(2,880) + (120) + 0 = 3,000$ . * The answer is correctly confirmed as $3,000$ .