2.6 A Pinch of History

From the Lalitavistara to the Googol

Indian and Buddhist thinkers were enchanted by colossal numbers long before anyone coined the phrase “scientific notation.” In the Lalitavistara, a Buddhist text dated to roughly the first century BCE, the figure Arjuna runs through number-names for the odd powers of ten, climbing all the way to 10^{53}: a hundred kotis make an ayuta (10^9), a hundred ayutas a niyuta (10^{11}), and the chain continues up to a tallakshana (10^{53}).

Mahaviracharya catalogued terms reaching 10^{23} in his Ganita-sara-sangraha, while one anonymous Jaina treatise supplied a name for every power of ten up to 10^{96}. The words we use daily echo the same scheme: a hundred thousand (10^2\times10^3 = 10^5) is a lakh, a hundred lakhs (10^7) a crore, a hundred crores (10^9) an arab, then a kharab (10^{11}), neel (10^{13}), padma (10^{15}), shankh (10^{17}).

The number 10^{100} wears a celebrated nickname, the googol, and 10^{\text{googol}} is a googolplex, dwarfing the estimated 10^{78} to 10^{82} atoms in the cosmos. The largest-denomination banknote ever issued was the one-sextillion (10^{21}) pengő note printed in Hungary in 1946.

Figure it Out: Mixed Practice

Practice

  1. Find the units digit of 3^{200}\div 9^{40}. (Hint: 9 = 3^2.)
  2. A crate holds 6 jars, and a fresh crate arrives daily. How many jars after 35 days?
  3. Judge each statement, Always True, Only Sometimes True, or Never True, with reasons:   (i) Cube numbers are also square numbers.   (ii) Fourth powers are also square numbers.   (iii) The fifth power of a number is divisible by the cube of that number.   (iv) The product of two cube numbers is a cube number.   (v) q^{46} is both a 4th power and a 6th power (q prime).
  4. Simplify in exponential form:   (i) 10^{-3}\times 10^{-4}   (ii) 6^9\div 6^5   (iii) 7^{-3}\div 7^5   (iv) (11^{-2})^{-4}   (v) m^4 n^7(mn)^6
  5. If 13^2 = 169, find:   (i) (1.3)^2   (ii) (0.13)^2   (iii) (0.013)^2   (iv) 130^2
  6. Circle the equal ones:   2^4\times 3^6,\quad 6^4\times 3^2,\quad 6^{10},\quad 18^2\times 6^2,\quad 6^{24}
  7. Identify the greater number:   (i) 5^3 or 3^5   (ii) 2^9 or 9^2   (iii) 50^2 or 2^{50}
  8. 64 is at once a square (8^2) and a cube (4^3). Are there other numbers that are both? Describe them in general.
  9. A digital vault uses a length-6 alphanumeric passcode (digits 0–9 and letters A–Z). How many such codes exist?
  10. World sheep population (2024) \approx 10^9, goats also \approx 10^9. What is the total?   (i) 20^9   (ii) 10^{11}   (iii) 10^{10}   (iv) 10^{18}   (v) 2\times10^9   (vi) 10^9 + 10^9
  11. A single bacterium divides into two every hour. Write, in exponential form, how many there are after 10 hours, and find the actual number.
  1. 3^{200}\div 9^{40} = 3^{200}\div 3^{80} = 3^{120}. The units digit of powers of 3 cycles 3,9,7,1 with period 4; since 120 is a multiple of 4, the units digit is 1.
  2. Each day adds 6 jars for 35 days: 6\times 35 = 210 = 2.1\times 10^2 jars.
    1. Only Sometimes True, holds exactly when the number is a 6th power, since n^6 = (n^2)^3 = (n^3)^2. (ii) Always True, n^4 = (n^2)^2. (iii) Always True, n^5 \div n^3 = n^2, a whole number. (iv) Always True, a^3\times b^3 = (ab)^3. (v) Never True, 46 is divisible by neither 4 nor 6.
    1. 10^{-7}   (ii) 6^4   (iii) 7^{-8}   (iv) 11^{8}   (v) m^{4+6}n^{7+6} = m^{10}n^{13}.
    1. 1.69   (ii) 0.0169   (iii) 0.000169   (iv) 16900.
  6. 2^4\times3^6 = 6^4\times3^2 = 18^2\times6^2 are equal (each = 11664). (6^{10} and 6^{24} are not.)
    1. 3^5 = 243 > 125 = 5^3   (ii) 2^9 = 512 > 81 = 9^2   (iii) 2^{50} is vastly greater than 50^2 = 2500.
  8. Yes, infinitely many. A number that is simultaneously a perfect square and a perfect cube is precisely a 6th power n^6: e.g. 1^6 = 1, 2^6 = 64, 3^6 = 729, 4^6 = 4096, …
  9. 36 choices per slot (26 letters + 10 digits), so 36\times36\times36\times36\times36\times36 = 36^6 = 2{,}17{,}67{,}82{,}336 codes.
  10. (vi) 10^9 + 10^9 = 2\times 10^9, which is also option (v); the total is 2\times 10^9.
  11. Each hour doubles, so after 10 hours there are 2^{10} = 1024 bacteria.