Memorise these
Squares 1 to 25. Non-negotiable for timed screens. In particular 13^2=169, 17^2=289, 19^2=361, 23^2=529.
Powers of 2 to 2^16: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536.
Cubes to 10: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
Square roots by bracketing
Find the two known squares either side, then interpolate.
sqrt(50): between 7^2=49 and 8^2=64. Much closer to 49, so about 7.07.
The interpolation rule: sqrt(a^2 + b) ≈ a + b/(2a).
sqrt(150): 12^2 = 144, remainder 6, so 12 + 6/24 = 12.25. (True: 12.247.)
Accurate to a few decimal places whenever b is small relative to a^2.
Squaring near a known value
Same binomial idea: (a + b)^2 = a^2 + 2ab + b^2.
52^2 = 2500 + 200 + 4 = 2704. 97^2 = 9409, from 10000 - 600 + 9.
Useful constants
- sqrt(2) ≈ 1.414, sqrt(3) ≈ 1.732, sqrt(5) ≈ 2.236, sqrt(10) ≈ 3.162
- e ≈ 2.718, pi ≈ 3.1416
- ln(2) ≈ 0.693, ln(10) ≈ 2.303
- log10(2) ≈ 0.301, log10(3) ≈ 0.477
Powers and logs
log10(2) ≈ 0.301 means 2^10 ≈ 10^3, which is why 1024 is close to 1000. That single fact lets you estimate large powers of 2 quickly: 2^20 ≈ 10^6, 2^30 ≈ 10^9.
For growth questions, the rule of 72 usually beats computing logs.
Where these appear
Volatility scaling uses sqrt(252) ≈ 15.9, which is worth knowing outright for Sharpe annualisation. Option and probability questions lean on sqrt(2) and sqrt(2 pi) ≈ 2.5.
Drill in mental math.