Mathematics

Scientific notation

• Example: $2.1 \times 10^2$
  • $2.1$ is the significand (also called coefficient or mantissa)
• Significand can’t begin with $0$

Significant figures

• Sig figs show measurement uncertainty
• Last digit is the least certain one
  • Example: $55.2$ → written as having 3 digits, but that last $2$ is the uncertain one in the measurement
• All digits between the first nonzero on the left and last nonzero on the right are significant
• Zeroes to the left of first nonzero digit = leading zeroes → not significant
• Zeroes to the right with a decimal point are significant
  • Example: $3400.0$ has $5$ sig figs

Multiplication and division with sig figs

• Answer should have same number of sig figs as the factor with the least sig figs

Exponents

• $X^A \times X^B = X^{A + B}$
• $\dfrac{X^A}{X^B} = X^{A - B}$
• $(X^A)^B = X^{A \times B}$
• $X^{-A} = \dfrac{1}{X^A}$
• $X^{A/B} = \sqrt[B]{X^A}$
• You can break up radicals:
  • $\sqrt{180} = \sqrt{4} \times \sqrt{9} \times \sqrt{5} = 2 \times 3 \times \sqrt{5} = 6\sqrt{5}$

Logarithms (base $A$ unless stated)

• $\log_A(1) = 0$
• $\log_A(A) = 1$
• $\log_A(A \times B) = \log_A(A) + \log_A(B)$
• $\log_A\left(\dfrac{A}{B}\right) = \log_A(A) - \log_A(B)$
• $\log_A(A^B) = B \log_A(A)$
• $\log_A\left(\dfrac{1}{A}\right) = -\log_A(A)$

“p” notation

• $p = -\log$
  • Example: $pH = -\log[H^+]$

Common vs. natural logs

• Common log base $$10$ → $\log$
• Natural log base $$e$ (about $2.718$) → $\ln$
• Rough relationship:
  • $\log(x) \approx \dfrac{\ln(x)}{2.3}$

• SOHCAHTOA:
  • $\sin = \dfrac{\text{opposite}}{\text{hypotenuse}}$
  • $\cos = \dfrac{\text{adjacent}}{\text{hypotenuse}}$
  • $\tan = \dfrac{\text{opposite}}{\text{adjacent}}$
• Ranges:
  • $\sin$ and $\cos$ range from $-1$ to $1$
  • $\tan$ ranges from $-\infty$ to $+\infty$

Unit conversions

• Fahrenheit to Celsius relation:
  • $F = \dfrac{9}{5} C + 32$

Solving algebraic systems

• Substitution
  • Solve one equation for a variable, substitute into the other
  • Example idea: solve for $y$ in terms of $x$, then plug into the second equation and solve
• Setting equations equal
  • If both equations are solved for the same variable (like $y$), set them equal and solve
• Elimination
  • Multiply one equation by a constant so terms line up
  • Add or subtract to eliminate one variable
  • Then solve for the remaining variable