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Chemistry · Ch 2 — Introduction to Analytical Chemistry

Scientific notation (exponential notation)

2.3.1

Scientific notation (exponential notation)

Chemists routinely have to work with numbers that are extremely large (for example, there are roughly 602,200,000,000,000,000,000,000 molecules in 2 g of hydrogen gas) or extremely small (for example, a single hydrogen atom has a mass of about 0.00000000000000000000000166 g). Writing out all the zeros in such numbers every time is impractical and error-prone, so scientific notation (also called exponential notation) is used instead. In scientific notation, any number is written in the form N×10nN \times 10^n, where the exponent nn can be a positive or negative whole number, and the coefficient NN is restricted to the range 1≤N<101 \leq N < 10. Using this form, the two numbers above become 6.022×10236.022 \times 10^{23} (molecules per 2 g of H2, closely related to Avogadro's number) and 1.66×10−241.66 \times 10^{-24} g. As another example, the ordinary number 123.546 becomes 1.23546×1021.23546 \times 10^2 in scientific notation — the decimal point has been moved two places to the left, and the exponent (2) records exactly how many places it moved. Likewise, a small number like 0.00015 is written 1.5×10−41.5 \times 10^{-4} — moving the decimal point four places to the right this time gives a negative exponent of −4-4. Once numbers are in this form, addition and subtraction req …

Misc Problem 2.1Adding numbers written in scientific notation

Worked out. To add two numbers in scientific notation, first rewrite them so both carry the same power of ten, then add the coefficients while keeping that common exponent. Worked example: add 5.55×1045.55 \times 10^4 and 6.95×1036.95 \times 10^3. Rewrite 6.95×1036.95 \times 10^3 with exponent 4: 6.95×103=0.695×1046.95 \times 10^3 = 0.695 \times 10^4. Now both terms share the exponent 4, so the coefficients can simply be added: 5.55×104+0.695×104=(5.55+0.695)×104=6.245×1045.55 \times 10^4 + 0.695 \times 10^4 = (5.55 + 0.695) \times 10^4 = 6.245 \times 10^4. The exponent itself is never added — only the coefficients …

Misc Problem 2.2Subtracting numbers written in scientific notation

Worked out. Subtraction in scientific notation follows the same rule as addition: equalise the exponents first, then subtract the coefficients. Worked example: subtract 5.8×10−35.8 \times 10^{-3} from 3.5×10−23.5 \times 10^{-2}. Rewrite 5.8×10−35.8 \times 10^{-3} with exponent −2-2: 5.8×10−3=0.58×10−25.8 \times 10^{-3} = 0.58 \times 10^{-2}. The subtraction becomes (3.5×10−2)−(0.58×10−2)=(3.5−0.58)×10−2=2.92×10−2(3.5 \times 10^{-2}) - (0.58 \times 10^{-2}) = (3.5 - 0.58) \times 10^{-2} = 2.92 \times 10^{-2}. As with addition, the power of ten only changes if the resulting coefficient falls outside the 1–10 range …

Misc Problem 2.3Multiplying numbers in scientific notation with positive exponents

Worked out. To multiply numbers in scientific notation, multiply the coefficients together and add the exponents of ten (the exponents are NOT multiplied). Worked example: (5.6×105)×(6.9×108)(5.6 \times 10^5) \times (6.9 \times 10^8). Multiply the coefficients: 5.6×6.9=38.645.6 \times 6.9 = 38.64. Add the exponents: 5+8=135 + 8 = 13. This gives 38.64×101338.64 \times 10^{13}, which is not yet in proper scientific notation since the coefficient exceeds 10. Moving the decimal one place left and increasing the exponent by one gives the final answer $3.864 \times …

Misc Problem 2.4Multiplying numbers in scientific notation with negative exponents

Worked out. The same multiplication rule applies when one or both exponents are negative: multiply the coefficients and algebraically add the exponents. Worked example: (9.8×10−2)×(2.5×10−6)(9.8 \times 10^{-2}) \times (2.5 \times 10^{-6}). Multiply the coefficients: 9.8×2.5=24.509.8 \times 2.5 = 24.50. Add the exponents algebraically: −2+(−6)=−8-2 + (-6) = -8. This gives 24.50×10−824.50 \times 10^{-8}, which needs renormalising because the coefficient is not between 1 and 10; moving the decimal one place left and increasing the exponent by one gives the final answer 2.45×10−72.45 \times 10^{-7}. In general, moving a decimal point n places to the right decreases the exponent by n, and moving it n places to th …

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