2.5 Have You Ever Wondered?
Linear growth set against exponential growth
Priya daydreams about a staircase climbing to the Moon. If every step rises 25 cm, how many steps reach the Moon, 3{,}84{,}400 km away? Dividing, we land on about 1{,}53{,}76{,}00{,}000 steps, roughly 1.5 billion. Each step adds a fixed 25 cm; this steady, additive ascent is linear growth.
Set the two journeys to the Moon side by side:
- Linear: 25 + 25 + 25 + \dots, about one and a half billion steps.
- Exponential: 0.001 \times 2 \times 2 \times 2 \times \dots, a mere 46 folds!
Two Flavours of Growth Linear growth is additive, you add the same amount at each step. Exponential growth is multiplicative, you multiply by the same factor at each step. Exponential growth dawdles at first, then surges past any linear growth, often by a staggering margin.
Building a feel for large numbers
A lakh is 10^5, a crore is 10^7, an arab is 10^9; a million is 10^6 and a billion is 10^9. Powers of ten let us line up the wildly mismatched sizes of things around us:
| Power of 10 | Roughly how many… |
|---|---|
| 10^2 | Kakapo birds alive (\approx 2\times10^2) |
| 10^5 | African elephants (\approx 4\times10^5) |
| 10^7 | camels worldwide (\approx 3.5\times10^7) |
| 10^9 | humans on Earth (\approx 8.2\times10^9) |
| 10^{12} | trees on Earth (\approx 3\times10^{12}) |
| 10^{16} | ants on Earth (\approx 2\times10^{16}) |
| 10^{23} | stars in the observable universe (\approx 2\times10^{23}) |
With about 8\times10^9 people and 4\times10^5 African elephants, there are roughly \dfrac{8\times10^9}{4\times10^5} = 2\times10^4 = 20{,}000 people per elephant.
Did You Know? Each rise of just one in the exponent makes a number ten times bigger. So the gap between 10^{16} ants and 10^{23} stars is not “a bit more”, it is a factor of 10^{7}, that is ten million. If every ant on Earth were a single star, you would still need ten million Earths’ worth of ants to match the stars in the observable universe. The exponent, not the coefficient, is where the real action lives.
Figure it Out: Powers of Ten in the World
Practice
Using world population \approx 8.2\times10^9, ants \approx 2\times10^{16}, trees \approx 3\times10^{12}, stars \approx 2\times10^{23}, give each answer in scientific notation:
- How many ants are there for every human?
- If a sparrow roost holds 20{,}000 birds and the world has about 1.6\times 10^9 sparrows, how many roosts could there be?
- If each tree bore about 10^4 leaves, how many leaves on all the trees?
- If one star were counted every second, how many seconds to count all the stars in the universe?
- \dfrac{2\times10^{16}}{8.2\times10^9} \approx 0.244\times10^{7} \approx 2.4\times10^{6} ants per human.
- \dfrac{1.6\times10^{9}}{2\times10^{4}} = 0.8\times10^{5} = 8\times10^{4} roosts.
- 3\times10^{12}\times 10^{4} = 3\times10^{16} leaves.
- One star per second means about 2\times10^{23} seconds.