Scientific Notation

The number of molecules in a single drop of water is roughly 1,670,000,000,000,000,000,000. Writing that out is absurd - and impossible to work with mentally. Scientific notation condenses it to 1.67 x $10^{21}$: a coefficient between 1 and 10 multiplied by a power of 10. This format makes enormous and minuscule numbers manageable, which is why every MCAT science passage uses it.

Standard Form

A number in scientific notation has the form N x $10^{n}$, where:

• N (the coefficient) is between 1 and 10
• n (the exponent) is a whole number (positive or negative)

Examples:

• 6,020,000,000,000,000,000,000,000 = 6.02 x $10^{23}$ (Avogadro’s number)
• 0.000000001 = 1 x $10^{-9}$ (one nanometer in meters)
• 0.00340 = 3.40 x $10^{-3}$

To convert a large number: move the decimal left until you have a number between 1 and 10. The number of places you moved is the positive exponent.

To convert a small number: move the decimal right until you have a number between 1 and 10. The number of places you moved is the negative exponent.

Multiplying in Scientific Notation

Multiply the coefficients, add the exponents.

Example: (3 x $10^{4}$) x (2 x $10^5$) = 6 x $10^{9}$

Example: (4 x $10^{3}$) x (5 x $10^{6}$) = 20 x $10^{9}$ = 2.0 x $10^{10}$ (adjusted)

Dividing in Scientific Notation

Divide the coefficients, subtract the exponents.

Example: (8 x $10^{7}$) / (2 x $10^{3}$) = 4 x $10^{4}$

Example: (3 x $10^5$) / (6 x $10^{8}$) = 0.5 x $10^{-3}$ = 5.0 x $10^{-4}$ (adjusted)

Adding and Subtracting in Scientific Notation

You must first make the exponents the same. Then add or subtract the coefficients.

Example: 3.0 x $10^{4}$ + 5.0 x $10^{3}$

Convert to the same exponent: 3.0 x $10^{4}$ + 0.5 x $10^{4}$ = 3.5 x $10^{4}$

Significant Figures

Significant figures indicate the precision of a measurement. The MCAT rarely penalizes for sig figs, but you should understand the basic rules:

Counting sig figs:

• All nonzero digits are significant: 345 has 3 sig figs
• Zeros between nonzero digits are significant: 305 has 3 sig figs
• Leading zeros are not significant: 0.0045 has 2 sig figs
• Trailing zeros after a decimal point are significant: 3.40 has 3 sig figs

Operations:

• Multiplication/Division: The result has the same number of sig figs as the measurement with the fewest sig figs
• Addition/Subtraction: The result has the same number of decimal places as the measurement with the fewest decimal places

In practice, MCAT answer choices are spaced far enough apart that sig figs rarely determine which answer you choose. Focus on getting the right order of magnitude and a close coefficient.

Order of Magnitude

An order of magnitude is the power of 10 closest to a number. This is useful for quick sanity checks.

• 500 is on the order of $10^{3}$ (closer to 1000 than to 100… actually closer to $10^{2.7}$, but we round)
• 3 x $10^{-8}$ is on the order of $10^{-7}$ to $10^{-8}$

If your answer is off by an order of magnitude (a factor of 10), something went wrong - usually a unit conversion error or a misplaced decimal.