4.2 Early Numeration Systems

Understand and Convert Babylonian Numerals to Hindu-Arabic Numerals

The Babylonians used a mix of an additive system of numbers and a positional system of numbers. An additive system is a number system where the value of repeated instances of a symbol is added the number of times the symbol appears. A positional system is a system of numbers that multiplies a “digit” by a number raised to a power, based on the position of the “digit.”

The Babylonian place values didn’t use powers of 10, but instead powers of 60. They didn’t use 60 different symbols though. For the value 1, they used the following symbol:

For values up to 9, that symbol would be repeated, so three would be written as

To represent the quantity 10, they used

For 20, 30, 40, and 50, they repeated the symbol for 10 however many times it was needed, so 40 would be written

When they reached 60, they moved to the next place value. The complete list of the Babylonian numerals up to 59 is in Table 4.1.

Table 4.1 Babylonian Numerals

You can see how Babylonians repeated the symbols to indicate multiples of a value. The number 6 is 6 of the symbol for 1 grouped together. The symbol for 30 is three of the symbols for 10 grouped together. However, their system doesn’t go past 59. To go past 59, they used place values. As opposed to the Hindu-Arabic system, which was based on powers of 10, the Babylonian positional system was based on powers of 60. You should also notice there is no symbol for 0, which has some impact on the number system. Since the Babylonian number system lacked a 0, they didn’t have a placeholder when a power of 60 was absent. Without a 0, 101, 110, and 11 all look the same. However, there is some evidence that the Babylonians left a small space between "digits" where we would use a 0, allowing them to represent the absence of that place value. To summarize, the Babylonian system of numbers used repeating a symbol to indicate more than one, used place values, and lacked a 0.

So how do we convert from Babylonian numbers to Hindu-Arabic numbers? To do so, we need to use the symbols from Table 4.1, and then place values based on powers of 60. If you have $n$ digits in the Babylonian number, you multiply the first “digit” by 60 raised to one less than the number of “digits.” You then continue through the “digits,” multiplying each by 60 raised to a power that is one smaller. However, be careful of spaces, since they represent a zero in that place.

Example 4.4

Converting Two-Digit Babylonian Numbers to Hindu-Arabic Numbers

Convert the Babylonian number

into a Hindu-Arabic number.

Solution

has two digits:

and

Step 1: So the first symbol,

represents 4 in the Babylonian system. This is multiplied by 60 to the first power (just as would happen in a two digit number), which gives us $4\times {60}^1$.

Step 2: The next symbol is

which represents 27 in the Babylonian system. This is multiplied by 60 raised to 0, which gives $4\times {60}^1+27\times {60}^0$.

Step 3: Calculating that yields $4\times {60}^1+27\times {60}^0=240+27=267$. So the Babylonian number

equals 267 in the Hindu-Arabic number system.

Your Turn 4.4

1.

Convert the Babylonian number into a Hindu-Arabic number.

Example 4.5

Converting Three-Digit Babylonian Numbers to Hindu-Arabic Numbers

Convert the Babylonian number

into a Hindu-Arabic number.

Solution

has three digits:

and

and

Step 1: So the first symbol,

represents 13 in the Babylonian system. This is multiplied by 60 to the second power (since there are 3 digits), which gives us $13\times {60}^2$.

Step 2: The next symbol is

which represents 8 in the Babylonian system, is multiplied by 60 raised to the first power, which gives us $13\times {60}^2+8\times {60}^1$.

Step 3: The last digit is

representing 54, which is multiplied by 60 raised to 0, which gives $13\times {60}^2+8\times {60}^1+54\times {60}^0$.

Step 4: Calculating that yields $13\times {60}^2+8\times {60}^1+54\times {60}^0=13\times \mathrm{3,600}+8\times 60+54\times 1=\mathrm{46,800}+480+54=\mathrm{47,334}$.

So, the Babylonian number

equals 47,334 in the Hindu-Arabic number system.

Your Turn 4.5

1.

Convert the Babylonian number into a Hindu-Arabic number.

Example 4.6

Converting Four-Digit Babylonian Numbers to Hindu-Arabic Numbers

Convert the Babylonian number

into a Hindu-Arabic number.

Solution

It appears that

has three digits, but there is a space in between

and

Remember, the Babylonian system has no 0, it instead employs a space where we expect a zero. This means this is a four digit number.

Step 1: The first symbol,

represents 12 in the Babylonian system. This is multiplied by 60 to the third power since there are four digits, which gives us $12\times {60}^3$.

Step 2: The next symbol is a blank, which for us is a 0, representing $0\times {10}^2$, giving us $12\times {60}^3+0\times {10}^2$.

Step 3: The next symbol is

which represents 42 in the Babylonian system, is multiplied by 60 raised to the first power, which gives us $12\times {60}^3+0\times {10}^2+42\times {60}^1$.

Step 4: The last Babylonian digit,

represents 39 in the Babylonian system. This is multiplied by 60 raised to 0, which gives $12\times {60}^3+0\times {10}^2+42\times {60}^1+39\times {60}^0$.

Step 5: Calculating that yields

$$\begin{array}{l}12\times {60}^3+0\times {10}^2+42\times {60}^1+39\times {60}^0\\ =12\times \mathrm{216,000}+0\times {10}^2+42\times 60+39\times 1\\ =\mathrm{2,592,000}+0+\mathrm{2,520}+39\\ =\mathrm{2,594,559}\end{array}$$

So the Babylonian number

equals 2,594,559 in the Hindu-Arabic number system.

Your Turn 4.6

1.

Convert the Babylonian number into a Hindu-Arabic number.

Understand and Convert Mayan Numerals to Hindu-Arabic Numerals

The Mayans employed a positional system just as we do and the Babylonians did, but they based their position values on powers of 20 and they had a dedicated symbol for zero. Similar to the Babylonians, the Mayans would repeat symbols to indicate certain values. A single dot was a 1, two dots were a 2, up to four dots. Then a five was a horizontal bar. The horizontal bars could be used three times, since the fourth horizontal bar would make a 20, which was a new position in the number. The 0 was a special picture, which appears like a turtle lying on its back. The shell would then be "empty," so maybe that’s why the symbol was 0. The complete list is provided in Table 4.2. Another feature of Mayan numbers was that they were written vertically. The powers of 20 increased from bottom to top.

Table 4.2 Mayan Numerals

To summarize, the Mayan system of numbers used repeating symbol to indicate more than one, used place values, and employed a 0. So how do we convert from Mayan numbers to Hindu-Arabic numbers? To do so, we need to use the symbols from Table 4.2 and then place values based on powers of 20. If you have $n$ digits in the Mayan number, you multiply the first “digit” by 20 raised to one less than the number of “digits.” You then continue through the “digits,” multiplying each by 20 raised to a power that is one smaller than the previous power. Fortunately, there is an explicit 0, so there is no ambiguity about numbers like 110, 101, and 11.

Example 4.7

Converting Two-Digit Mayan Numbers to Hindu-Arabic Numbers

Convert the Mayan number

into a Hindu-Arabic number.

Solution

has two digits:

and

Step 1: So, the first symbol,

represents 15 in the Mayan system. This is multiplied by 20 to the first power, which gives us $15\times {20}^1$.

Step 2: The next symbol is

which represents 9 in the Mayan system. This is multiplied by 20 raised to 0, which gives $15\times {20}^1+9\times {20}^0$.

Step 3: Calculating that yields $15\times {20}^1+9\times {20}^0=300+9=309$. So

equals 309 in the Hindu-Arabic number system.

Your Turn 4.7

1.

Convert the Mayan number into a Hindu-Arabic number.

Example 4.8

Converting Three-Digit Mayan Numbers to Hindu-Arabic Numbers

Convert the Mayan number

into a Hindu-Arabic number.

Solution

has three digits:

and

and

Step 1: So the first symbol,

represents 6 in the Mayan system. This is multiplied by 20 to the second power (since there are 3 digits), which gives us $6\times {20}^2$.

Step 2: The next symbol is

which represents 8 in the Mayan system, is multiplied by 20 raised to the first power, which gives us $6\times {20}^2+8\times {20}^1$.

Step 3: The last digit is

representing 4, which is multiplied by 20 raised to 0, which gives $6\times {20}^2+8\times {20}^1+4\times {20}^0$.

Step 4: Calculating that yields $6\times {20}^2+8\times {20}^1+4\times {20}^0=6\times 400+8\times 20+4\times 1=\mathrm{2,400}+160+4=\mathrm{2,564}$. So the Mayan number

equals 2,564 in the Hindu-Arabic number system.

Your Turn 4.8

1.

Convert the Mayan number into a Hindu-Arabic number.

Example 4.9

Converting Four-Digit Mayan Numbers to Hindu-Arabic Numbers

Convert the Mayan number

into a Hindu-Arabic number.

Solution

has four digits, so the first power of 20 that is used is 3.

Step 1: The first symbol,

represents 8 in the Mayan system. This is multiplied by 20 to the third power (since there are four digits), which gives us $8\times {20}^3$.

Step 2: The next symbol is

which is a 0, representing $0\times {20}^2$, giving us $8\times {20}^3+0\times {20}^2$.

Step 3: The next symbol is

which represents 16 in the Mayan system, is multiplied by 20 raised to the first power, which gives us $8\times {20}^3+0\times {20}^2+16\times {20}^1$.

Step 4: The last Mayan digit,

represents 5 in the Mayan system. This is multiplied by 20 raised to 0, which gives $8\times {20}^3+0\times {20}^2+16\times {20}^1+5\times {20}^0$.

Step 5: Calculating that yields

$$\begin{array}{l}8\times {20}^3+0\times {20}^2+16\times {20}^1+5\times {20}^0\\ =8\times 8000+0\times 400+16\times 20+5\times 1\\ =\mathrm{64,000}+0+320+5\\ =\mathrm{64,325}\end{array}$$

So the Mayan number

equals 64,325 in the Hindu-Arabic number system.

Your Turn 4.9

1.

Convert the Mayan number into a Hindu-Arabic number.

Understand and Convert Between Roman Numerals and Hindu-Arabic Numerals

The Mayan and Babylonian systems shared two features, one of which we are familiar with (place value) and one that we don’t use (repeated symbols). The Roman system of numbers used repeated symbols, but does not employ a place value. It also lacks a 0. The Roman system is built on the following symbols in Table 4.3.

As in the Mayan and Babylonian systems, a symbol may be repeated to indicate a larger value. However, at 4, they did not use IIII. They instead used IV. Since the I came before the V, the number stands for “one before five.” A similar process was used for 9, which was written IX, or “one before ten.” The value 40 was written XL, or “ten before fifty,” while 49 was written XLIX, or “forty plus nine.”

The following are the rules for writing and reading Roman numerals.

• The representations for bigger values precede those for smaller values.
• Up to three symbols may be grouped together; for example, III for 3, or XXX for 30, or CC for 200.
• A larger value followed by a smaller value indicated addition; for example, VII for 7, XIII for 13, LV for 55, and MCC for 1200.
• I can be placed before V to indicate 4, or before X, to indicate 9. These are the only ways I is used as a subtraction.
• X can be placed before L to indicate 40, and before C to indicate 90. These are the only ways X is used as a subtraction.
• C can be placed before D to indicate 400, and before M to indicate 900. These are the only ways C is used as a subtraction.
• If multiple symbols are used, and a subtraction involving that symbol, the subtraction part comes after the multiple symbols. For example, XXIX for 29 and CCXC for 290.

Of course, we can convert from Hindu-Arabic numerals, to Roman numerals, too.