Scientific notation: a × 10^n with 1 ≤ |a| < 10

The rule

A number is in scientific notation if written as:

$a \times 10^n$

with:

• $1 \leq |a| < 10$ (one nonzero digit before the decimal)
• $n$ an integer

How to convert to scientific notation

Large number: move the decimal to the left until 1 digit remains before it.

$45{,}600 = 4.56 \times 10^4$ (decimal moved 4 places).

Small number: move the decimal to the right, $n$ becomes negative.

$0.00072 = 7.2 \times 10^{-4}$ (decimal moved 4 places right).

Order of magnitude

Once written $a \times 10^n$, the power $10^n$ gives the order of magnitude.

• $4.56 \times 10^4$ ⇒ order $10^4$ (tens of thousands).
• $7.2 \times 10^{-4}$ ⇒ order $10^{-4}$ (ten-thousandths).

Multiplication in scientific notation

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

Then adjust to stay in scientific notation if $a \times b \geq 10$.

Example: $(3 \times 10^4) \times (5 \times 10^2) = 15 \times 10^6 = 1.5 \times 10^7$.

Division

$\dfrac{a \times 10^n}{b \times 10^m} = \dfrac{a}{b} \times 10^{n-m}$

Common mistakes

1. Leaving $a \geq 10$ or $a < 1$ (not in scientific form).
2. Wrong sign of $n$ (left = positive, right = negative).
3. Counting zeros instead of decimal-move places.
4. Confusing $10^n$ with $n \times 10$.

?À toi de jouer

Scientific notation of 302,000?

A.3.02 × 10⁵B.302 × 10³C.3.02 × 10⁶D.30.2 × 10⁴

Valider