Math Talk
Q1. How will you use sticks to answer the other two questions (Q2 and Q3)?
Answer:
To answer Q2 (whether we have fewer cows than our neighbour), we can create a one-to-one mapping between sticks and cows for our herd, and similarly for the neighbour's herd. By comparing the number of sticks in each collection, we can see which collection has fewer sticks, indicating fewer cows. To answer Q3 (how many more cows we need to have the same number), we can count the difference in sticks; if our collection has fewer sticks, we need that many more cows.
Q2. How many numbers can you represent in this way using the sounds of the letters of your language?
Answer:
This depends on the language. For example, if the language has 26 letters (like English), you can represent numbers 1 to 26 using single letters. However, if you form combinations (like 'aa' for 27), you can represent infinitely many numbers by creating a systematic list of sounds corresponding to letters and their combinations, following a fixed order.
Q3. Do you see a way of extending this method to represent bigger numbers as well? How?
Answer:
Yes, the Roman number system (Table 1) can be extended by introducing additional symbols for larger values, such as L (50), C (100), D (500), and M (1000). Numbers can be combined by adding or subtracting these symbols (e.g., IV = 4, IX = 9, XL = 40, XC = 90) to represent larger numbers. However, this system still has limitations for very large numbers.
Figure it Out
Q1. Suppose you are using the number system that uses sticks to represent numbers, as in Method 1. Without using either the number names or the numerals of the Hindu number system, give a method for adding, subtracting, multiplying and dividing two numbers or two collections of sticks.
Answer:
For addition: Combine the two collections of sticks into one pile and count the total by making a one-to-one mapping with sticks. For subtraction: Remove sticks equal to the smaller collection from the larger collection, and the remaining sticks indicate the difference. For multiplication: Start with an empty collection, repeatedly add sticks equal to one collection as many times as there are sticks in the other collection, then count the total. For division: Repeatedly remove sticks equal to the divisor from the dividend until no sticks are left or a remainder remains; the number of times you removed is the quotient.
Q2. One way of extending the number system in Method 2 is by using strings with more than one letter—for example, we could use 'aa' for 27. How can you extend this system to represent all the numbers? There are many ways of doing it!
Answer:
One method is to treat each letter as a digit in a base-26 system. For numbers 1-26, assign letters 'a' to 'z'. For 27, combine two letters: 'aa' (where the first letter represents groups of 26 and the second represents units). For 28, use 'ab', and so on. Continue this pattern, adding more letters for larger numbers, similar to how the decimal system uses digits 0-9.
Q3. Try making your own number system.
Answer:
Students can create their own number system. For example, they can choose symbols for base 5: use 0 (circle), 1 (line), 2 (cross), 3 (triangle), 4 (square). For larger numbers, combine symbols place-value style, e.g., 5 = 'line circle' (one group of five and zero ones). They should also define rules for operations and counting.
Q1. Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707.
Answer:
To represent numbers in the Egyptian system, we need to break them into powers of 10 (1, 10, 100, 1000, etc.) and use the corresponding symbols. Since the symbols are not provided in the text, the representation should follow the pattern described: for 10458 = 10000+400+50+8, we would use the symbol for 10000 once, symbol for 100 four times, symbol for 10 five times, and symbol for 1 eight times. Similarly for other numbers.
Q2. What numbers do these numerals stand for?
Answer:
The text does not provide specific numerals to decode. Without the actual symbols shown in the textbook, we cannot determine the numbers. Students should refer to the symbols table in the textbook to decode.
Q1. Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.
Answer:
In base-5 system, landmark numbers are powers of 5: 1, 5, 25, 125, 625,...
- 15 = 2×5 + 5×1 = two 5s and five 1s
- 50 = 2×25 + 0×5 + 0×1 = two 25s
- 137 = 1×125 + 0×25 + 2×5 + 2×1 = one 125, two 5s, two 1s
- 293 = 2×125 + 1×25 + 3×5 + 3×1 = two 125s, one 25, three 5s, three 1s
- 651 = 1×625 + 0×125 + 2×25 + 0×5 + 1×1 = one 625, two 25s, one 1
(Actual symbols should be used from Table 2)
Q2. Is there a number that cannot be represented in our base-5 system above? Why or why not?
Answer:
No, there is no natural number that cannot be represented. Any natural number can be expressed as a sum of powers of 5 multiplied by coefficients from 0 to 4. This is because base-5 is a complete number system - any positive integer has a unique representation in base-5 using the digits 0-4.
Q3. Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?
Answer:
Landmark numbers of base-7: 7^0=1, 7^1=7, 7^2=49, 7^3=343, 7^4=2401, 7^5=16807, ...
In general, landmark numbers of a base-n system are: n^0=1, n^1=n, n^2, n^3, n^4, ... (all powers of n starting from n^0)
Q1(i). Add the following Egyptian numerals: (i) [symbols not shown in text]
Answer:
Without the actual symbols provided in the textbook, we cannot perform the addition. The process would involve: count all symbols of each type, group 10 smaller symbols to form one larger symbol, continue until no group of 10 exists.
Q1(ii). Add the following Egyptian numerals: (ii) [symbols not shown in text]
Answer:
Same as above - requires the actual symbols from the textbook.
Q2. Add the following numerals that are in the base-5 system that we created: [symbols not shown in text]
Answer:
Without the actual symbols, we can explain: In base-5, whenever we have 5 of a landmark number, they combine to form the next higher landmark number. The addition process involves counting all symbols, grouping 5 of same type to make one of the next type, and repeating.
Q1. Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?
Answer:
No. Because the Egyptian system is a base-10 system where 10 of any landmark number makes the next higher landmark number. If any symbol appears 10 or more times, we would regroup 10 of them to form one of the next higher symbol. This is the fundamental rule of the system - you always group 10 collections to get the next landmark number.
Q2. Create your own number system of base 4, and represent numbers from 1 to 16.
Answer:
In base-4, landmark numbers are powers of 4: 1, 4, 16. We need symbols for these (students can invent their own).
Representations:
1 = 1
2 = 2
3 = 3
4 = 1×4 + 0×1
5 = 1×4 + 1×1
6 = 1×4 + 2×1
7 = 1×4 + 3×1
8 = 2×4 + 0×1
9 = 2×4 + 1×1
10 = 2×4 + 2×1
11 = 2×4 + 3×1
12 = 3×4 + 0×1
13 = 3×4 + 1×1
14 = 3×4 + 2×1
15 = 3×4 + 3×1
16 = 1×16 + 0×4 + 0×1
Q3. Give a simple rule to multiply a given number by 5 in the base-5 system that we created.
Answer:
In base-5 system, multiplying any number by 5 is equivalent to shifting all symbols to the next higher landmark number. Since 5 times a landmark number gives the next landmark number, each symbol moves one position higher. For example, 5 × (some representation) just replaces each symbol with the symbol for the next higher power of 5.
Q1. Represent the following numbers in the Mesopotamian system — (i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605
Answer:
In Mesopotamian base-60 system, we use symbols for 1 and 10 (as mentioned in text). Numbers are grouped into powers of 60.
(i) 63 = 1×60 + 3×1 → symbols: one group of 60 (using symbol for 1 in 60s position), then three 1s
(ii) 132 = 2×60 + 12×1 → two 60s, then 12 (which is 10+1+1 using the 10 symbol and two 1 symbols)
(iii) 200 = 3×60 + 20×1 → three 60s, then 20 (two 10 symbols)
(iv) 60 = 1×60 + 0×1 → one 60, blank for ones (requires placeholder)
(v) 3605 = 1×3600 + 0×60 + 5×1 → one 3600 (60²), blank for 60s, five 1s
Try This
Q1. Quickly count the number of objects in each of the following boxes: Up to what group size could you immediately see the number of objects without counting?
Answer:
This is an activity where the student looks at boxes with different numbers of dots or objects. Most humans can immediately recognise the number of objects up to 4 or 5 without counting (a phenomenon called subitising). For larger groups, counting becomes necessary. The student should identify the group size they can perceive at a glance.