Scientific Notation

Solve problems involving very small and very large measurements, time scales and intervals expressed in scientific notation.

One number between 1 and 10, times a power of ten. It turns unreadable numbers into ones you can compare at a glance.

Method

The standard form

a \times 10^n \quad \text{where } 1 \leq a < 10

Exactly one digit before the point.

Large numbers take a positive power

4\,500\,000 = 4{,}5 \times 10^6

Count how many places the point moves left.

Small numbers take a negative power

0{,}00032 = 3{,}2 \times 10^{-4}

Count how many places it moves right.

Multiplying: add the powers

(a \times 10^m)(b \times 10^n) = ab \times 10^{m+n}

Then fix $a$ back into range if needed.

Comparing is easy: check the power first

Only compare the front numbers if the powers match.

e.g.

\text{Write } 0{,}000\,067 \text{ in scientific notation, then multiply it by } 3 \times 10^{9}.

Move the point so one non-zero digit sits in front. $6{,}7$
Count how many places you moved it. Five, to the right. $\text{right} \Rightarrow \text{negative power}$
Write the standard form. $6{,}7 \times 10^{-5}$
To multiply, handle the front numbers and the powers separately. $6{,}7 \times 3 = 20{,}1$
Add the powers. $10^{-5} \times 10^{9} = 10^{4}$
Fix the front number back into range. $20{,}1 \times 10^4 = 2{,}01 \times 10^5$

One digit before the point. Right means negative, left means positive.

\text{Write } 92\,000\,000 \text{ in scientific notation.}

Answer

9{,}2 \times 10^7

\text{Write } 0{,}0045 \text{ in scientific notation.}

Answer

4{,}5 \times 10^{-3}

\text{Write } 3{,}6 \times 10^{-2} \text{ as an ordinary number.}

Answer

0{,}036

(2 \times 10^5)(4 \times 10^3)

Answer

8 \times 10^8

\text{Which is larger: } 3 \times 10^6 \text{ or } 9 \times 10^5?

Answer

3 \times 10^6

\text{Why is } 12 \times 10^4 \text{ not standard form?}

Answer

\text{The front number must be under 10.}