49.2: Introduction

Last updated
Save as PDF

Page ID: 29401

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Scientific notation indicates the number of 10’s by which a coefficient is multiplied or divided. A negative exponent indicates the coefficient is divided by 10’s, which means the decimal would move to the left for each negative power. A positive exponent indicates the coefficient is multiplied by 10’s, which means the decimal would move to the right for each positive power. Numbers in the physical sciences can be very large or very small, making it cumbersome to write the entire number in standard notation, so powers of ten are used as a shorthand. When two numbers with powers of 10 are multiplied, the coefficients are multiplied, and then the powers of 10 exponents are added. When two numbers with powers of 10 are divided, the coefficients are divided, and then the powers of 10 exponents are subtracted.

The Meaning of Powers of 10

Table \(\PageIndex{1}\)
Standard	Scientific	Powers of Ten
\(540,000,000,000\)	\(5.4 \times 10^{11}\)	\(5.4 \times (10\times 10\times 10\times 10\times 10\times 10\times 10\times 10\times 10\times 10\times 10)\)
\(0.00000101\)	\(1.01 \times 10^{-6}\)	\(\dfrac{1.01}{(10\times 10\times 10\times 10\times 10\times 10)}\)

Table \(\PageIndex{2}\)
Standard	Scientific	Decimal Places
\(540,000,000,000\)	\(5.4 \times 10^{11}\)	move decimal 11 places to the right
\(0.00000101\)	\(1.01 \times 10^{-6}\)	move decimal 6 places to the left

Examples

Table \(\PageIndex{3}\): Multiply & Divide Powers of 10
\(10^6 \times 10^2 = 10^8\)	exponents: 6 + 2 = 8
\(10^6 \times 10^{-2} = 10^4\)	exponents: 6 + (−2) = 4
\(10^6 \div 10^2 = 10^4\)	exponents: 6 – 2 = 4
\(10^6 \div 10^{-2} = 10^8\)	exponents: 6 – (−2) = 8

Table \(\PageIndex{4}\): Multiply & Divide with Coefficients
\((2 \times 10^3) \times (4 \times 10^5)\)	\((2 \times 4) \times 10^{3+5} = 8 \times 10^8\)
\((2 \times 10^3) \div (4 \times 10^5)\)	\((2 \div 4) \times 10^{3-5} = 0.5 \times 10^{-2}\)

Contributors and Attributions

Template:ContribCCPhySc101L