Base-10:
Our counting system is a base-10 system. This means that our basic counting structure is based on collecting groups of 10. Our place values in our numbers (ones, tens, hundreds, etc.) are essentially the way we keep track of our groups of 10 as we count. We have the numbers 0-9. When we are counting and reach the number 10, we are basically saying that we have 1 group of 10; as we keep counting, 20 means that we have 2 groups of 10. Continuing on, 25, for example, means 2 groups of 10 with 5 individual pieces. If we look closely at our place values, we find:
ones= digits 0-9 (the parts of a group of 10) [this can also be looked at as the number of individual parts multiplied by 100]
tens= how many groups of 10 [also the number of groups multiplied by 101]
hundreds= how many 10 groups of 10 [also the number of groups multiplied by 102]
thousands= how many 100 groups of 10 [also the number of groups multiplied by 103]
etc.
So, thinking about it in this way, the number 532,749 means that we have:
9 singles
4 groups of 10
7 groups of 10 tens
2 groups of 100 tens
3 groups of 1,000 tens
5 groups of 10,000 tens
Other Bases:
The concept of other bases means that rather than bundling groups of 10, we bundle groups of other numbers. Clocks, for example, work on base-12. Every 12 hours we complete a full "bundle". Thinking in base-12, the numbers 0-9, 10 (represented as T), and 11 (represented as E) would go in the "ones" column, and the number 10 would actually mean that you have 1 full bundle of 12.
Some examples in base-12:
*the number 20 would mean that you have 2 full bundles of 12 (or 24 in our base-10 system)
*the number 25 would mean that you have 2 full bundles of 12, plus 5 more from a partial bundle (or 24+5, or 29 in our base-10 system)
*a trickier one- the number 3E would mean that you have 3 full bundles of 12, plus 11 more from a partial bundle (or 36+11, or 47 in our base-10 system)
What does each place value mean in the base-12 system?
"Ones"= "digits" 0-12 [or the number of individual pieces multiplied by 120]
"Tens" (or Twelves)= how many groups of 12 [or the number of groups multiplied by 121]
"Hundreds" (or 144's)= how many 12 groups of 12 [or the number of groups multiplied by 122]
"Thousands" (or 1,728's)= how many 144 groups of 12 [or the number of groups multiplied by 123]
Binary, or base-2, is often associated with computers. In addition to our 12 hour clock system, base-24 could also be used when dealing with hours.
Parsha Connection:
In this weeks parsha, we read about the shmita year, which occurs every 7 years once the Jewish people entered into the land of Israel. We then learn that after 7 cycles of 7 years there is a yovel year. Here we see a system for counting in groups of 7- basically we are calculating in base-7.
How can we use our base notation to keep track of the years for shmita and yovel?
Let's think about this. First of all, let's set-up what each place value is representing.
"Ones"= digits 0-6 [or the number of individual pieces multiplied by 70]
"Tens" (or Sevens)= how many groups of 7 [or the number of groups multiplied by 71]
"Hundreds" (or 49's)= how many 7 groups of 7 [or the number of groups multiplied by 72]
"Thousands" (or 343's)= how many 49 groups of 7 [or the number of groups multiplied by 73]
Let's try to map out the years for shmita and yovel using base-7 notation:
Regular Counting year
|
Base-7 notation
|
Special Year
|
Year 1
|
1
|
|
Year 2
|
2
|
|
Year 3
|
3
|
|
Year 4
|
4
|
|
Year 5
|
5
|
|
Year 6
|
6
|
|
Year 7
|
10
|
Shmita
|
Year 8
|
11
|
|
Year 9
|
12
|
|
Year 10
|
13
|
|
Year 11
|
14
|
|
Year 12
|
15
|
|
Year 13
|
16
|
|
Year 14
|
20
|
Shmita
|
Year 15
|
21
|
|
Year 16
|
22
|
|
Year 17
|
23
|
|
Year 18
|
24
|
|
Year 19
|
25
|
|
Year 20
|
26
|
|
Year 21
|
30
|
Shmita
|
Year 22
|
31
|
|
Year 23
|
32
|
|
Year 24
|
33
|
|
Year 25
|
34
|
|
Year 26
|
35
|
|
Year 27
|
36
|
|
Year 28
|
40
|
Shmita
|
Year 29
|
41
|
|
Year 30
|
42
|
|
Year 31
|
43
|
|
Year 32
|
44
|
|
Year 33
|
45
|
|
Year 34
|
46
|
|
Year 35
|
50
|
Shmita
|
Year 36
|
51
|
|
Year 37
|
52
|
|
Year 38
|
53
|
|
Year 39
|
54
|
|
Year 40
|
55
|
|
Year 41
|
56
|
|
Year 42
|
60
|
Shmita
|
Year 43
|
61
|
|
Year 44
|
62
|
|
Year 45
|
63
|
|
Year 46
|
64
|
|
Year 47
|
65
|
|
Year 48
|
66
|
|
Year 49
|
70
|
Shmita
|
Year 50
|
71
|
Yovel
|
Year 51
|
72
|
What patterns do you notice here?
If you continue the pattern, you'll notice a slight shift over time. Can you find a pattern in the way that the first pattern shifts?
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