Powers of 10 and scientific notation

You already write numbers in expanded form: 47561 = (4 × 10000) + (7 × 1000) + (5 × 100) + (6 × 10) + 1. With powers of 10 that becomes (4 × 10⁴) + (7 × 10³) + (5 × 10²) + (6 × 10¹) + (1 × 10⁰).

Negative powers carry the same idea past the decimal point: 561.903 = (5 × 10²) + (6 × 10¹) + (1 × 10⁰) + (9 × 10⁻¹) + (0 × 10⁻²) + (3 × 10⁻³). Each place is one power of 10 smaller than the one before.

Now try reading these: the Sun is 30,00,00,00,00,00,00,00,00,000 m from the centre of the Milky Way; our galaxy has 1,00,00,00,00,000 stars; the Earth's mass is 59,76,00,00,00,00,00,00,00,00,000 kg. Hardly anyone reads them correctly. 'We may miscount the number of zeroes or place commas incorrectly... It is like getting ₹5,000 when you were supposed to get ₹50,000.'

Any number can be written as a number from 1 up to (but not including) 10, times a power of 10: 5900 = 5.9 × 10³, 20800 = 2.08 × 10⁴, 80,00,000 = 8 × 10⁶.

That's scientific notation, also called standard form: x × 10ʸ, where the coefficient x is at least 1 and less than 10, and the exponent y is any integer. (24 × 10³ is not standard form: 24 is too big. Write it 2.4 × 10⁴.)

Here's why the exponent matters most. Mumbai's population of about 2 crore is 2 × 10⁷. Change the 2 to a 3 and it grows by half, from 2 crore to 3 crore. Change the 7 to an 8 and it becomes 20 crore: ten times as many. The small number up top does the heavy lifting.

The coefficient tells you something else: how well we know the figure. If Kohima's population is given as 1,42,395, it suggests we're sure right down to the units. If we only know it's about 1 lakh 42 thousand, we write 1.42 × 10⁵; if only about 1 lakh 40 thousand, 1.4 × 10⁵.

So the number of digits in the coefficient is a claim about accuracy. These values are 'rounded-off estimates, averages, or approximations; most of the time, they serve the purpose at hand.'