Standard form calculator

Standard form (scientific notation) is useful for numbers that would otherwise get lost among the zeros. Instead of 384,400,000 metres to the Moon, we write 3.844 × 10⁸ m, and you can see straight away how many orders of magnitude that is.

Going the other way, from standard form back to an ordinary number: enter the coefficient and the power. The distance from the Earth to the Sun, 1.496 × 10⁸ km, is 149,600,000 km.

The calculator still writes very large numbers (from 10¹⁵) and very small ones (below 10⁻⁶) with a power of ten, because nobody could keep track of the zeros.

How it works

A number is written as a product

$$x=a\times {10}^n,1\le |a|<10$$

where a is a number with a single non-zero digit before the decimal point, and n is an integer, the power of ten. Large numbers have a positive n, and small numbers between zero and one have a negative one.

We move the decimal point until only one non-zero digit is left in front of it. The exponent n tells us how many places, and in which direction, we moved it. For 384,400,000 the decimal point is at the end, and we move it eight places to the left:

$$\mathrm{384,400,000}=3.844\times {10}^8$$

For a small number the point moves to the right and the exponent is negative: 0.00052 = 5.2 × 10⁻⁴. The exponent is, incidentally, the integer part of the base-10 logarithm of |x|, which is how the calculator works it out.