Mathematics from zero
Linear vs exponential
Two savings plans. One adds 10 coins to your jar every day. The other doubles whatever is in your jar every day. Start them both at the same time. For a while the first looks better — but give it ten days, and the second is not just ahead, it is out of sight.
After this lesson you can tell linear growth from exponential growth, compute a few steps of each, and explain why exponential growth always overtakes linear growth in the end.
Linear growth adds the same amount each step. This is the steady climb from the functions unit: every step of input, the total rises by a fixed amount. Add 10 a day and the jar holds 10, 20, 30, 40 — a constant step of 10. The growth never speeds up.
Exponential growth multiplies by the same amount each step. Instead of adding a fixed amount, exponential growth multiplies by a fixed amount. Double each day starting from 1, and the jar holds 1, 2, 4, 8, 16. The step itself keeps growing, because each step acts on a bigger number than the last.
Exponential growth starts slow, then overtakes any linear growth. Early on, adding 10 a day beats doubling from 1 — on day 2 the linear jar has 20, the doubling jar only 4. But doubling compounds: by day 7 it has passed 100, and from then on linear growth cannot catch up. Multiplying always wins the long race against adding.
This contrast shapes the real world. Money left to earn interest, a population of animals, a rumour spreading person to person — all grow by multiplying, not adding, so all grow exponentially. Recognising whether something adds or multiplies tells you, in advance, whether it will creep or explode.
Plan A adds 5 each day, starting from 0. Plan B doubles each day, starting from 1. Compare them day by day to day 7.
Plan A (linear, add 5): day 1 is 5, day 2 is 10, then 15, 20, 25, 30, day 7 is 35.
Plan B (exponential, double): day 1 is 2, day 2 is 4, then 8, 16, 32, 64, day 7 is 128.
For the first three days Plan A is ahead — on day 3, A has 15 and B has only 8. But B crosses over: by day 4 they are close, and by day 7 B’s 128 dwarfs A’s 35. The longer you wait, the wider the gap.
Why this works
Why does exponential growth always overtake linear growth, no matter how big the linear step is? Because a linear step is fixed forever, but an exponential step grows with the total. Even a tiny multiplier — like 1.01 — eventually produces steps larger than any fixed number, because the thing being multiplied keeps getting bigger. Fixed beats growing only for a while; growing wins in the end.
Common mistake
A common mistake is judging growth by its early days. In the first few steps linear growth often looks faster, and people conclude it is the bigger plan. It is not — it is just ahead for now. To compare growth types, look at the long run, not the first few steps, because that is where multiplying pulls away.
Linear growth: start at 0, add 10 each day. What is the total after 3 days? Type it.
Exponential growth: start at 1, double each day. What is the total after 3 days? Type it.
Linear growth: start at 0, add 4 each day. What is the total after 5 days? Type it.
Exponential growth: start at 1, multiply by 3 each step. What is the total after 2 steps? Type it.
Over the long run, which grows faster? Type 1 for exponential, 0 for linear.
What is the difference between linear and exponential growth?
Linear growth adds the same fixed amount each step; exponential growth multiplies by the same fixed amount each step. Exponential growth often starts slower, but because its step grows with the total, it always overtakes linear growth in the long run. Money, populations, and spreading rumours grow by multiplying — so to predict whether something will creep or explode, ask whether it adds or multiplies.