1.1 Scientific notationIB Maths: Applications and Interpretation HL: Revision notes

A number is in scientific notation (standard form) when it is written as $a\times10^{k}$ with $1\le a<10$ and $k$ an integer. A large number has a positive exponent $k$ and a number between $0$ and $1$ has a negative exponent. Examples: $149\,600\,000=1.496\times10^{8}$ and $0.000\,072=7.2\times10^{-5}$. Count how many places the decimal point moves. Numbers such as $45\times10^{3}$ or $0.45\times10^{5}$ equal $4.5\times10^{4}$ but are not in standard form, because $a$ is not between 1 and 10.

Multiply or divide the numbers $a$, and add or subtract the exponents: $(a\times10^{m})(b\times10^{n})=ab\times10^{m+n},\qquad\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m-n}.$ Then check that the number is between 1 and 10 and adjust the exponent if it is not. Example: $(3.2\times10^{6})(5\times10^{-9})=16\times10^{-3}=1.6\times10^{-2}$. Example: $\frac{4.8\times10^{15}}{3.2\times10^{6}}=1.5\times10^{9}$.

You cannot add or subtract the numbers directly unless the powers of ten match. First write both numbers with the same power of ten, then add or subtract the numbers and write the answer in standard form. Example: $5.97\times10^{24}+7.35\times10^{22}=5.97\times10^{24}+0.0735\times10^{24}=6.0435\times10^{24}\approx6.04\times10^{24}$. Example: $3.2\times10^{4}-5\times10^{3}=3.2\times10^{4}-0.5\times10^{4}=2.7\times10^{4}$.

Changing units changes the power of ten. Going to a smaller unit gives a bigger number, so the exponent goes up; going to a larger unit makes it go down. 1 km $=10^{3}$ m, 1 m $=10^{3}$ mm, 1 kg $=10^{3}$ g, 1 day $=8.64\times10^{4}$ s. Example: a mass of $5.97\times10^{24}$ kg is $5.97\times10^{24}\times10^{3}=5.97\times10^{27}$ g. A length of $7.5\times10^{-6}$ m is $7.5\times10^{-6}\times10^{3}=7.5\times10^{-3}$ mm. Write units correctly, for example m s$^{-1}$, m$^{2}$ and cm$^{3}$.

Your GDC shows large numbers as, for example, 5.2E30. This is calculator notation and is not acceptable in an answer: write $5.2\times10^{30}$. Enter standard form with the exponent key (usually EE or $\times10^{x}$) and use brackets around the whole of a denominator, for example $4.8\times10^{15}\div(3.2\times10^{6})$. The IB default is to give answers exactly or to 3 significant figures. Estimate first as a check: $\frac{1.99\times10^{30}}{5.97\times10^{24}}\approx\frac{2}{6}\times10^{6}=0.33\times10^{6}$, so the answer should be about $3.3\times10^{5}$.