2.5: Scientific Notation
- Page ID
- 541104
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Write very large or small numbers using scientific notation
- Enter numbers in Scientific Notation into your calculator correctly
One of the most important numbers in chemistry is 602,200,000,000,000,000,000,000. This number, named after, but not defined by Amedeo Avogadro, is referred to by chemists as "1 mole". (Chemists use it the way bakers use "1 dozen".) It's very difficult to write and work with numbers this large, so we use a system called scientific notation.
A number in Scientific notation takes the form: N x 10ⁿ
So Avogadro's number written in scientific notation is 6.022 x 10²³
The first step to writing numbers in scientific notation is to recall how exponents of ten work.
10¹ = 10
10² = 100
10³ = 1,000
10⁴ = 10,000
The second step is to remember how multiplication by powers of ten works.
1.5 x 10 = 15 or 1.5 x 101 = 15
1.5 x 100 = 150 or 1.5 x 10² = 150
1.5 x 1000 = 1,500 or 1.5 x 10³ = 1,500
1.5 x 10000 = 15,000 or 1.5 x 10⁴ = 15,000
It works the same with numbers smaller than 1, we just use negative exponents.
10⁻¹ = 0.1
10⁻² = 0.01
10⁻³ = 0.001
10⁻⁴ = 0.0001
and
1.5 x 10⁻¹ = 0.15
1.5 x 10⁻² = 0.015
1.5 x 10⁻³ = 0.0015
1.5 x 10⁻⁴ = 0.00015
Putting Numbers in Scientific Notation
- Count over until the decimal is right behind the first non-zero digit,
- then use positive exponents for numbers bigger than 1 or negative exponents for numbers smaller than 1 (decimals).
| Decimal Format | Scientific Notation |
|---|---|
| 274 | 2.74 × 102 |
| 0.0035 | 3.5 × 10–3 |
| 60221415 | 6.0221415 × 107 |
| 0.125 | 1.25 × 10–1 |
| 402.5 | 4.025 × 102 |
| 0.0002001 | 2.001 × 10–4 |
| 10,000 | 1 × 104 |
Convert the following numbers into scientific notation:
- 93,000,000
- 708,000
- 0.000248
- 0.073
- Answer a
-
9.3 x 10⁷
- Answer b
-
7.08 x 10⁵
- Answer c
-
2.48 x 10⁻⁴
- Answer d
-
7.3 x 10⁻²
Convert the following numbers from scientific notation into decimal format:
- 6.02 × 104
- 6.31 × 10-4
- 4.68 × 10-2
- 9.3 × 107
- Answer a
-
60200
- Answer b
-
0.000631
- Answer c
-
0.0468
- Answer d
-
93000000