Uncertainty in Measurement and Scientific Notation

Chemistry constantly asks you to work with numbers that are staggeringly large or vanishingly small. Consider this: just 2 grams of hydrogen gas contains roughly 602,200,000,000,000,000,000,000 molecules. A single hydrogen atom weighs about 0.00000000000000000000000166 grams. Other fundamental constants, such as Planck’s constant, the speed of light, and the charge on an electron, all involve numbers of similar magnitude. Writing out all those zeros every time you need to multiply, divide, or compare would be impractical and error-prone.

This topic introduces a clean, efficient way to handle such numbers. But first, let’s understand why the accuracy of the measurements behind these numbers matters, and who is responsible for keeping that accuracy in check.

How Reference Standards Keep Measurements Trustworthy

Every measuring instrument in a laboratory, from a simple metre stick to a high-precision analytical balance, is only as reliable as the standard it has been calibrated (adjusted to read correctly) against. After scientists agree on a unit of measurement, such as the kilogram or the metre, they also agree on a reference standard that all devices worldwide can be compared to. Without this shared anchor, measurements taken in different labs would not be comparable.

The Kilogram Standard

Since 1889, every mass measurement in the world has traced back to a single physical object: a cylinder crafted from platinum-iridium ($\text{Pt-Ir}$) alloy, sealed inside an airtight jar at the International Bureau of Weights and Measures in Sevres, France. This cylinder was the definition of the kilogram. Why $\text{Pt-Ir}$? Because this alloy barely reacts with anything. Its extreme resistance to chemical attack means the cylinder’s mass stays effectively unchanged over centuries.

Relying on one physical artefact to anchor a fundamental unit is not ideal, though. What if the cylinder picks up a few atoms of contamination, or loses a few through handling? For this reason, scientists have been working to redefine the kilogram using a constant of nature: the Avogadro constant. The approach is to count, with extreme precision, the number of atoms in a carefully prepared sample of known mass. One method uses X-rays to map the atomic spacing in a crystal of ultrapure silicon, achieving an accuracy of about 1 part in $10^6$. At the time this textbook was written, no method had proven reliable enough to officially replace the $\text{Pt-Ir}$ cylinder, though a change was expected within the decade.

The Metre Standard and Its Evolution

The metre has undergone several redefinitions as measurement technology improved:

• Original definition — the length between two marks on a $\text{Pt-Ir}$ bar maintained at a temperature of $0°\text{C}$ ($273.15 \text{ K}$).
• 1960 redefinition — the metre was set equal to $1.65076373 \times 10^6$ times the wavelength of light emitted by a krypton laser. This was an awkward number, but it preserved the metre at its previously agreed length while tying it to an atomic property rather than a physical bar.
• 1983 redefinition (current) — the CGPM defined the metre as the length of the path travelled by light in vacuum during a time interval of $\frac{1}{299\,792\,458}$ of a second. This links the metre directly to the speed of light.

Similar reference standards exist for every other physical quantity (time, temperature, electric current, and so on), each anchored to a fundamental constant of nature.

Who Maintains These Standards: National Metrology Institutes

Definitions of units are not set in stone. As experimental methods advance and more precise ways of realising a unit become available, the member nations of the Metre Treaty (an international agreement dating back to 1875) collectively agree to update the formal definition. This ongoing process is how the metre moved from a physical bar to a laser wavelength, and eventually to the speed of light.

Putting these definitions into practice on the ground requires dedicated institutions. Every modern industrialised country maintains a National Metrology Institute (NMI) for this purpose. India’s NMI is the National Physical Laboratory (NPL), based in New Delhi, and it carries out three essential tasks:

• Realising units — it conducts experiments to physically establish both base units and derived units of measurement
• Maintaining national standards — it keeps India’s official measurement standards
• International comparison — it periodically cross-checks its standards against those held by NMIs in other countries and by the International Bureau of Standards in Paris

This global network of comparison ensures that a kilogram measured in India matches a kilogram measured anywhere else in the world.

Scientific Notation: A Compact Way to Write Extreme Numbers

Now, back to the practical challenge. Try picking two of those enormous numbers mentioned at the start and adding them together. Better yet, try multiplying them by hand. You will quickly appreciate why chemists needed a better system for representing and calculating with such values.

The solution is scientific notation (also called exponential notation). In this system, any number is written in the form:

$N \times 10^n$

where:

• $N$ is the digit term, a number that lies between $1.000...$ and $9.999...$
• $n$ is the exponent, which can be positive or negative

The exponent tells you exactly how many places the decimal point has been shifted from the number’s original position.

Converting a Number to Scientific Notation

When the original number is greater than 10, you move the decimal to the left. Each position you shift adds $+1$ to the exponent.

Take $232.508$. Move the decimal two places to the left to bring the digit term into the 1-to-10 range:

$232.508 = 2.32508 \times 10^2$

The exponent is $+2$ because the decimal moved two places to the left.

When the original number is less than 1, you move the decimal to the right. Each position you shift adds $-1$ to the exponent.

Take $0.00016$. Move the decimal four places to the right:

$0.00016 = 1.6 \times 10^{-4}$

The exponent is $-4$ because the decimal moved four places to the right.

With this notation, even the most extreme numbers become compact and easy to read. And, as we will see, doing arithmetic with them becomes far simpler too.

Multiplying and Dividing in Scientific Notation

For multiplication and division, you handle the digit terms and the powers of 10 separately. The rules follow directly from the standard laws of exponents.

Multiplication

Multiply the digit terms together and add the exponents:

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

Worked Example 1:

$(5.6 \times 10^5) \times (6.9 \times 10^8)$

Step 1:Multiply the digit terms:

$5.6 \times 6.9 = 38.64$

Step 2:Add the exponents:

$5 + 8 = 13$

Step 3:Combine:

$38.64 \times 10^{13}$

Step 4:The digit term 38.64 is outside the 1-to-10 range, so adjust. Rewrite $38.64$ as $3.864 \times 10^1$, then combine with the existing power of 10:

$3.864 \times 10^1 \times 10^{13} = 3.864 \times 10^{14}$

Worked Example 2:

$(9.8 \times 10^{-2}) \times (2.5 \times 10^{-6})$

Step 1:Multiply digit terms:

$9.8 \times 2.5 = 24.50$

Step 2:Add exponents:

$(-2) + (-6) = -8$

Step 3:Combine:

$24.50 \times 10^{-8}$

Step 4:Adjust: $24.50 = 2.450 \times 10^1$, so:

$2.450 \times 10^{1} \times 10^{-8} = 2.450 \times 10^{-7}$

Division

Divide the digit terms and subtract the denominator’s exponent from the numerator’s exponent:

$\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$

Worked Example 3:

$\frac{2.7 \times 10^{-3}}{5.5 \times 10^4}$

Step 1:Divide digit terms:

$\frac{2.7}{5.5} = 0.4909$

Step 2:Subtract exponents:

$(-3) - 4 = -7$

Step 3:Combine:

$0.4909 \times 10^{-7}$

Step 4:Adjust: $0.4909 = 4.909 \times 10^{-1}$, so:

$4.909 \times 10^{-1} \times 10^{-7} = 4.909 \times 10^{-8}$

Adding and Subtracting in Scientific Notation

Addition and subtraction follow a different rule from multiplication and division. You cannot simply add or subtract the digit terms unless both numbers already share the same exponent. The first step is always to rewrite one of the numbers so that both exponents match. After that, you add or subtract the digit terms normally.

Addition

Worked Example 4: Add $6.65 \times 10^4$ and $8.95 \times 10^3$.

Step 1:The exponents are different ($10^4$ vs. $10^3$). Rewrite the smaller number using the larger exponent. To go from $10^3$ to $10^4$, shift the decimal one place to the left:

$8.95 \times 10^3 = 0.895 \times 10^4$

Step 2:Both numbers now have the same exponent ($10^4$), so add the digit terms:

$(6.65 \times 10^4) + (0.895 \times 10^4) = (6.65 + 0.895) \times 10^4$

$= 7.545 \times 10^4$

The digit term 7.545 is already between 1 and 10, so no further adjustment is needed.

Subtraction

Worked Example 5: Subtract $4.8 \times 10^{-3}$ from $2.5 \times 10^{-2}$.

Step 1:Match the exponents. Convert $4.8 \times 10^{-3}$ to the $10^{-2}$ scale by moving the decimal one place to the left:

$4.8 \times 10^{-3} = 0.48 \times 10^{-2}$

Step 2:Subtract the digit terms:

$(2.5 \times 10^{-2}) - (0.48 \times 10^{-2}) = (2.5 - 0.48) \times 10^{-2}$

$= 2.02 \times 10^{-2}$