Historical Counting Systems: Get Stronger

Counting Board And Quipu

1)      In the following Peruvian counting board, determine how many of each item is represented. Please show all of your calculations along with some kind of explanation of how you got your answer. Note the key at the bottom of the drawing.

Simple depiction of a Peruvian counting board. The key at the bottom shows that black pebbles represent jars, spotted pebbles represent baskets, and striped pebbles represent tools. There are three black pebbles, four spotted pebbles, and four striped pebbles in the outer square regions. There are no black pebbles, one spotted pebble, and one striped pebble in the larger white rectangular compartments. There are three black pebbles, four spotted pebbles, and two striped pebbles in the middle region. There are two black pebbles, four spotted pebbles, and two striped pebbles on the second levels. Finally, there is one black pebble, one spotted pebble, and two striped pebbles on the highest corner levels.

 

2)  Draw a quipu with a main cord that has branches (H cords) that show each of the following numbers on them. (You should produce one drawing for this problem with the cord for part a on the left and moving to the right for parts b through d.)

(a) [latex]232[/latex]

(b) [latex]5065[/latex]

(c) [latex]23,451[/latex]

(d) [latex]3002[/latex]

Basic Base Conversions

3) 423 in base 5 to base 10                4) 3044 in base 5 to base 10

5) 387 in base 10 to base 5                6) 2546 in base 10 to base 5

7) 110101 in base 2 to base 10               8) 11010001 in base 2 to base 10

9) 100 in base 10 to base 2                 10) 2933 in base 10 to base 2

11) Convert 653 in base 7 to base 10            12) Convert 653 in base 10 to base 7

13) 3412 in base 5 to base 2                14) 10011011 in base 2 to base 5

Mayan Conversions

Convert the following numbers to Mayan notation. Show your calculations used to get your answers.

 

15)      [latex]135[/latex]                                                                                 16) [latex]234[/latex]

 

17)      [latex]360[/latex]                                                                                 18) [latex]1,215[/latex]

 

19)      [latex]10,500[/latex]                                                                            20) [latex]1,100,000[/latex]

 

Convert the following Mayan numbers to decimal (base–[latex]10[/latex]) numbers. Show all calculations.

21)

three compartments stacked vertically - from the bottom: two horizontal lines and two dots, two dots, one dot

22)

three compartments stacked vertically - from the bottom: one shell, four dots, one dot

23)

 three compartments - from the bottom: three dots, one shell, three dots

 24)

4 compartments with mayan symbols - from the bottom: one shell, next, one shell, two horizontal lines stacked, two horizontal lines and two dots stacked.

James Bidwell has suggested that Mayan addition was done by “simply combining bars and dots and carrying to the next higher place.” He goes on to say, “After the combining of dots and bars, the second step is to exchange every five dots for one bar in the same position.” After converting the following base [latex]10[/latex] numbers into vertical Maya notation (in base [latex]20[/latex], of course), perform the indicated addition:

25)      [latex]32[/latex] + [latex]11[/latex]                                                                           26) [latex]82 + 15[/latex]

 

27)      [latex]35 + 148[/latex]                                                                      28) [latex]2412 + 5000[/latex]

 

29)      [latex]450 + 844[/latex]                                                             30) [latex]10,000 + 20,000[/latex]

 

31)      [latex]4,500 + 3,500[/latex]                                                      32) [latex]130,000 + 30,000[/latex]

 

33)         Use the fact that the Mayans had a base-[latex]20[/latex] number system to complete the following multiplication table. The table entries should be in Mayan notation. Remember: Their zero looked like this…. Xerox and then cut out the table below, fill it in, and paste it onto your homework assignment if you do not want to duplicate the table with a ruler.

(To think about but not write up: Bidwell claims that only these entries are needed for “Mayan multiplication.” What does he mean?)

 

x

 one dot  two dots  three dots  four dots  one horizontal line  two horizontal lines stacked  three horizontal lines stacked
one dot              
two dots              
three dots              
four dots              
one horizontal line              
two horizontal lines stacked              
three horizontal lines stacked