Question 2 - Figure it out - Page 80 - Chapter 3 Class 8 - A Story of Numbers (Ganita Prakash)

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Question 2 Form a base-2 place value system using ‘ukasar’ and ‘urapon’ as the digits. Compare this system with that of the Gumulgal’s.This question asks to create a binary (Base-2) system using the words from the Gumulgal counting system introduced earlier in the chapter. Digits: In Base-2, we need two digits (0 and 1). Let's assign: 0: (We need a symbol for zero, let's assume a placeholder like "void" or reuse a concept, but strictly speaking, urapon and ukasar correspond to 1 and 2 in the original text. For a binary system digits are usually 0 and 1. If we must use these two words as the digits): Let urapon = 0 Let ukasar = 1 (Alternatively, if we treat them as values 1 and 2, it's not a standard Base-2 system). Assumption: Let's map urapon to 0 and ukasar to 1 for a standard positional binary system. Counting in this Base-2 System: 0: urapon 1: ukasar 2 (10 in binary): ukasar-urapon 3 (11 in binary): ukasar-ukasar 4 (100 in binary): ukasar-urapon-urapon Comparison with Gumulgal's System: Gumulgal: Their system was additive (1, 2, 2+1, 2+2...). It was somewhat base-2 based, but not positional. "ukasar-ukasar" meant 2 + 2 = 4 Our Positional Base-2: "ukasar-ukasar" would mean 1 × 21 + 1 × 20 = 2 + 1 = 3 Difference: The Gumulgal system adds the values of the words together. A positional system uses the position of the word to multiply its value by a power of the base (1, 2, 4, 8...). (Alternatively, if we treat them as values 1 and 2, it's not a standard Base-2 system). Assumption: Let's map urapon to 0 and ukasar to 1 for a standard positional binary system. Counting in this Base-2 System: 0: urapon 1: ukasar 2 (10 in binary): ukasar-urapon 3 (11 in binary): ukasar-ukasar 4 (100 in binary): ukasar-urapon-urapon Comparison with Gumulgal's System: Gumulgal: Their system was additive (1, 2, 2+1, 2+2...). It was somewhat base-2 based, but not positional. "ukasar-ukasar" meant 2 + 2 = 4 Our Positional Base-2: "ukasar-ukasar" would mean 1 × 21 + 1 × 20 = 2 + 1 = 3 Difference: The Gumulgal system adds the values of the words together. A positional system uses the position of the word to multiply its value by a power of the base (1, 2, 4, 8...). Let’s look at some examples Comparing 1 in Both System Base-2 (Positional) vs. Gumulgal (Additive) New Base-2 System (Positional: Value depends on place) Representation: ukasar 2. Gumulgal System (Additive: Count by 2 s ) urapon Representation: uraponBase-2 (Positional) vs. Gumulgal (Additive) Select a Number (1-31): 2 Positional (New) (1) Additive (Old) New Base-2 System (Positional: Value depends on place) Representation: ukasar-urapon 2. Gumulgal System (Additive: Count by 2 s ) ukasar 2 Representation: ukasarComparing 3 in Both System teacho Base-2 (Positional) vs. Gumulgal (Additive) Select a Number (1-31): 3 Positional (New) Additive (Old) New Base-2 System (Positional: Value depends on place) Representation: ukasar-ukasar 2. Gumulgal System (Additive: Count by 2 s ) ukasar urapon Representation: ukasar-uraponComparing 4 in Both System teac Base-2 (Positional) vs. Gumulgal (Additive) New Base-2 System (Positional: Value depends on place) Representation: ukasar-urapon-urapon 2. Gumulgal System (Additive: Count by 2s) Representation: ukasar-ukasarComparing 10 in Both System tea Base-2 (Positional) vs. Gumulgal (Additive) Select a Number (1-31): 10 Positional (New) Additive (Old) New Base-2 System (Positional: Value depends on place) 16s 8s 4s 2s 1s urapon 0 ukasar 1 urapon 0 ukasar 1 urapon 0 Representation: ukasar-urapon-ukasar-urapon 2. Gumulgal System (Additive: Count by 2 s ) Representation: ukasar-ukasar-ukasar-ukasar-ukasarComparing 15 in Both System tea Base-2 (Positional) vs. Gumulgal (Additive) Select a Number (1-31): 15 Positional (New) Additive (Old) New Base-2 System (Positional: Value depends on place) Representation: ukasar-ukasar-ukasar-ukasar 2. Gumulgal System (Additive: Count by 2 s ) Representation: ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasaruraponComparing 16 in Both System teach Base-2 (Positional) vs. Gumulgal (Additive) Select a Number (1-31): 16 New Base-2 System (Positional: Value depends on place) Representation: ukasar-urapon-urapon-urapon-urapon 2. Gumulgal System (Additive: Count by 2 s ) Representation: ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasarukasarComparing 18 in Both System teace Base-2 (Positional) vs. Gumulgal (Additive) New Base-2 System (Positional: Value depends on place) Representation: ukasar-urapon-urapon-ukasar-urapon 2. Gumulgal System (Additive: Count by 2 s ) Representation: ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar