Scientific Notation

Scientific notation exists because counting zeros is slow and error-prone. Instead of writing 0.000000035, you write 3.5 times 10⁻⁸: one nonzero digit before the decimal point, times ten raised to whatever power captures the size of the number. That single digit rule is what makes the form "proper" - 35 times 10⁶ is correct in value but not in form, since 35 has two digits in front. Fix it by sliding the decimal left one place, which turns 35 into 3.5, and every place you slide left you pay for by raising the exponent by one - so 10⁶ becomes 10⁷, giving 3.5 times 10^7.

The mechanism behind multiplying or dividing numbers in this form is to treat the front numbers and the powers of ten separately. Multiply the front numbers together, add the exponents. Divide the front numbers, subtract the exponents. So (4.0 times 10⁵) times (2.5 times 10⁻²) becomes (4.0 times 2.5) times 10^{5 + (-2)}, which is 10.0 times 10^3. But 10.0 isn't between 1 and 10, so you slide the decimal left once more and bump the exponent up, landing on 1.0 times 10^4.

This topic sits next to work on exponent rules and radicals, since the same add-and-subtract logic for exponents shows up everywhere powers of ten appear. On the number and quantity items, expect a quick multiplication or division dressed up with variables standing in for digits, testing whether you can keep the front-number arithmetic and exponent arithmetic from blurring together.