Scientific notation

Picture trying to text someone the distance from Earth to the nearest star: 40 000 000 000 000 km. You'd miscount the zeros, they'd miscount the zeros, and someone ends up thinking it's 40 billion instead of 40 trillion. That's the exact problem scientific notation (also called standard form) was invented to fix.

Every number in scientific notation is written in the form $a\times 10^n$, where $a$ is the coefficient , a number satisfying $1\le a<10$ , and $n$ is an integer exponent that controls the scale. The coefficient holds all the meaningful digits; the power of ten tells you how big or small the number actually is.

Scientific Notation: A number expressed as $a\times 10^n$, where $1\le a<10$, $a\in \mathbb{R}$, and $n\in \mathbb{Z}$.

Here's the counterintuitive part: a negative exponent doesn't mean a negative number. $10^{-3}=\frac{1}{1000}=0.001$ , it just means the number is tiny, not below zero. Plenty of students mix these up on assessments, so lock that in now.