Leave lesson

Number · Standard form

1 / 9

Writing numbers in standard form

Why standard form fixes exactly one non-zero digit before the point, what the power of 10 really counts, why negative powers mean small (not negative), and how to compare sizes at a glance.

Number · Standard form

Writing numbers in standard form

Why standard form fixes exactly one non-zero digit before the point, what the power of 10 really counts, why negative powers mean small (not negative), and how to compare sizes at a glance.

Why it works

Standard form writes every number as

a×10n,1≤a<10,a \times 10^n, \qquad 1 \le a < 10,

— a number between 1 and 10, times a power of ten. The power does the heavy lifting: it says how many places the point moves, which is the number's size; the aa carries the digits. 45 000 000=4.5×10745\,000\,000 = 4.5 \times 10^7: start at 4.54.5 and push the point seven places right.

Why aa must sit between 1 and 10. Without that rule the same number has many costumes — 45×10645 \times 10^6, 4.5×1074.5 \times 10^7 and 0.45×1080.45 \times 10^8 are all equal — and none of them could be compared at a glance. Forcing one non-zero digit before the point makes the form unique, so 45×10645 \times 10^6 is not standard form: trade a factor of 10 from the 45 into the power, 4.5×1074.5 \times 10^7. (Coefficient down ÷10 ⇒ power up +1, and vice versa — the trade must balance.)

Negative powers mean small, never negative. 10−3=1100010^{-3} = \frac{1}{1000}, so 6.2×10−4=0.000626.2 \times 10^{-4} = 0.00062 — the point moves four places left. The number is still positive; the minus lives in the index, not the value. Count carefully: in 0.000620.00062 the point moves past three zeros and the 6 — four places.

Comparing: powers first, digits second. 1.2×1051.2 \times 10^5 beats 9.8×1039.8 \times 10^3 — no matter how big the digits look, an extra power of ten is a whole extra place. Only when powers tie do the coefficients decide. For small numbers remember the scale runs the other way: 10−610^{-6} is bigger than 10−910^{-9} (millionths beat billionths).

Reading it back. 3.08×1053.08 \times 10^5: move the point five places right, filling with zeros — 308 000308\,000. The zero inside 3.08 is a real digit and must survive the journey.