Algebra

Last modified: 2026-09-28 23:45:45 (PDT)

Mastery of Elementary Algebra (a.k.a. “College Algebra”) is a prerequisite for calculus, which is in turn a prerequisite for most statistics and data science courses. Nevertheless, each year, some students are still uncomfortable with algebraic manipulations of mathematical formulas. Therefore, I include this section as a quick reference.

1 Equalities

Theorem 1 (Equalities are transitive) If \(a=b\) and \(b=c\), then \(a=c\)

Theorem 2 (Substituting equivalent expressions) If \(a = b\), then for any function \(f(x)\), \(f(a) = f(b)\)

2 Inequalities

Theorem 3 (Adding to both sides of an inequality) If \(a<b\), then \(a+c < b+c\)

Theorem 4 (negating both sides of an inequality) If \(a < b\), then: \(-a > -b\)

Theorem 5 (Multiplying both sides of an inequality by a positive number) If \(a < b\) and \(c > 0\), then \(ca < cb\).

Theorem 6 (Negation is multiplication by \(-1\)) \[-a = (-1)*a\]

3 Infimum and supremum

Definition 1 (Infimum (greatest lower bound)) Let \(A \subseteq \mathbb{R}\) be nonempty and bounded below, meaning that some \(t \in \mathbb{R}\) satisfies \(t \le a\) for all \(a \in A\). The infimum of \(A\), written \(\inf A\), is the greatest real number \(t\) satisfying \(t \le a\) for all \(a \in A\):

\[\inf A \stackrel{\text{def}}{=}\max\mathopen{}\left\{t \in \mathbb{R}: \forall a \in A,\ t \le a\right\}\mathclose{}\]

If \(A\) is nonempty but not bounded below, we write \(\inf A = -\infty\) by convention.

The maximum in Definition 1 always exists: that is the completeness (greatest-lower-bound) property of the real numbers (Rudin 1976, Definition 1.8, p. 4, and Theorem 1.19, p. 8). If the infimum belongs to \(A\), it equals the minimum: \(\inf A = \min A\).

Example 1 (Numerical examples of infimum)  

• \(\inf\{1, 2, 3\} = 1\), since \(1\) is the smallest element.
• \(\inf(0.5, 1] = 0.5 = \min[0.5, 1]\): for intervals open below, the infimum equals the minimum of the corresponding closed-below interval, even though \(0.5 \notin (0.5, 1]\). More generally, \(\inf(c, b] = \min[c, b] = c\) for any \(c < b\).
• \(\inf\{t \ge 0 : t > 0.5\} = 0.5\), even though \(0.5\) itself is not in the set.
• \(\inf\{-1, -2, -3, \ldots\} = -\infty\), because no real number is less than or equal to every element of that set.

Definition 2 (Supremum (least upper bound)) Let \(A \subseteq \mathbb{R}\) be nonempty and bounded above, meaning that some \(t \in \mathbb{R}\) satisfies \(a \le t\) for all \(a \in A\). The supremum of \(A\), written \(\sup A\), is the smallest real number \(t\) satisfying \(a \le t\) for all \(a \in A\):

\[\sup A \stackrel{\text{def}}{=}\min\mathopen{}\left\{t \in \mathbb{R}: \forall a \in A,\ a \le t\right\}\mathclose{}\]

If \(A\) is nonempty but not bounded above, we write \(\sup A = +\infty\) by convention.

The minimum in Definition 2 always exists: that is the completeness (least-upper-bound) property of the real numbers (Rudin 1976, Definition 1.8, p. 4, and Theorem 1.19, p. 8). If the supremum belongs to \(A\), it equals the maximum: \(\sup A = \max A\).

Example 2 (Numerical examples of supremum)  

• \(\sup\{1, 2, 3\} = 3\), since \(3\) is the largest element.
• \(\sup\{t \ge 0 : t < 0.5\} = 0.5\), even though \(0.5\) itself is not in the set.
• \(\sup\{1, 2, 3, \ldots\} = +\infty\), because no real number is greater than or equal to every element of that set.

4 Sums

Theorem 7 (adding zero changes nothing) \[a+0=a\]

Theorem 8 (Sums are symmetric) \[a+b = b+a\]

Theorem 9 (Sums are associative)  

When summing three or more terms, the order in which you sum them does not matter:

\[(a + b) + c = a + (b + c)\]

5 Products

Theorem 10 (Multiplying by 1 changes nothing) \[a \times 1 = a\]

Theorem 11 (Products are symmetric) \[a \times b = b \times a\]

Theorem 12 (Products are associative) \[(a \times b) \times c = a \times (b \times c)\]

6 Division

Theorem 13 (Division can be written as a product) If \(b \neq 0\), then

\[\frac {a}{b} = a \times \frac{1}{b}\]

7 Sums and products together

Theorem 14 (Multiplication is distributive) \[a(b+c) = ab + ac\]

8 Quotients

Definition 3 (Quotient) For real numbers \(a\) and \(b\) with \(b \neq 0\), the quotient of \(a\) by \(b\) is the result of dividing \(a\) by \(b\):

\[\frac{a}{b}\]

A quotient is also called a fraction; \(a\) is its numerator and \(b\) its denominator. A quotient whose denominator measures time or population size is often called a rate; in epidemiology, rates typically have such a denominator.

cf. https://en.wikipedia.org/wiki/Rate_(mathematics)

Example 3 (A quotient) The quotient of \(6\) by \(4\) is \(\frac{6}{4} = 1.5\). The quotient of \(6\) by \(0\) is undefined, because Definition 3 requires a nonzero denominator.

Definition 4 (Ratios) A ratio is a quotient in which the numerator and denominator are measured using the same unit scales.

cf. https://en.wikipedia.org/wiki/Ratio

Definition 5 (Proportion) In statistics, a proportion typically means a ratio where the numerator represents a subset of the denominator.

See https://en.wikipedia.org/wiki/Population_proportion.

See also https://en.wikipedia.org/wiki/Proportion_(mathematics) for other meanings.

Definition 6 (Proportional) Two functions \(f(x)\) and \(g(x)\) are proportional if their ratio \(\frac{f(x)}{g(x)}\) does not depend on \(x\). (cf. https://en.wikipedia.org/wiki/Proportionality_(mathematics))

Additional reference for elementary algebra: https://en.wikipedia.org/wiki/Population_proportion#Mathematical_definition

9 Exponentials and Logarithms

In these notes, \(\operatorname{log}\mathopen{}\left\{x\right\}\mathclose{}\) is the natural logarithm of \(x > 0\), the logarithm with base \(e \approx 2.718\), and \(\operatorname{exp}\mathopen{}\left\{x\right\}\mathclose{} = e^x\) is the exponential function. Some sources write \(\ln x\) for the natural logarithm and reserve \(\log x\) for base 10.

Theorem 15 (\(\operatorname{exp}\mathopen{}\left\{\right\}\mathclose{}\) and \(\operatorname{log}\mathopen{}\left\{\right\}\mathclose{}\) are mutual inverses)  

1. For every \(a > 0\): \(\operatorname{exp}\mathopen{}\left\{\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}\right\}\mathclose{} = a\).
2. For every \(a \in \mathbb{R}\): \(\operatorname{log}\mathopen{}\left\{\operatorname{exp}\mathopen{}\left\{a\right\}\mathclose{}\right\}\mathclose{} = a\).

Theorem 16 (Logarithm of a product) If \(a > 0\) and \(b > 0\), then

\[ \operatorname{log}\mathopen{}\left\{a \cdot b\right\}\mathclose{} = \operatorname{log}\mathopen{}\left\{a\right\}\mathclose{} + \operatorname{log}\mathopen{}\left\{b\right\}\mathclose{} \]

Corollary 1 (Logarithm of a quotient) If \(a > 0\) and \(b > 0\), then

\[\operatorname{log}\mathopen{}\left\{\frac{a}{b}\right\}\mathclose{} = \operatorname{log}\mathopen{}\left\{a\right\}\mathclose{} - \operatorname{log}\mathopen{}\left\{b\right\}\mathclose{}\]

Proof. Since \(a > 0\) and \(b > 0\), the quotient \(\frac{a}{b}\) is positive, so Theorem 16 applies to the product \(\frac{a}{b} \cdot b\):

\[ \begin{aligned} \operatorname{log}\mathopen{}\left\{a\right\}\mathclose{} &= \operatorname{log}\mathopen{}\left\{\frac{a}{b} \cdot b\right\}\mathclose{} && \text{(} a = \tfrac{a}{b} \cdot b \text{)} \\ &= \operatorname{log}\mathopen{}\left\{\frac{a}{b}\right\}\mathclose{} + \operatorname{log}\mathopen{}\left\{b\right\}\mathclose{} && \text{(logarithm of a product)} \end{aligned} \]

The second step applies Theorem 16. Subtracting \(\operatorname{log}\mathopen{}\left\{b\right\}\mathclose{}\) from both sides gives \(\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{} - \operatorname{log}\mathopen{}\left\{b\right\}\mathclose{} = \operatorname{log}\mathopen{}\left\{\frac{a}{b}\right\}\mathclose{}\).

Theorem 17 (Logarithm of a power) If \(a > 0\) and \(b \in \mathbb{R}\), then

\[ \operatorname{log}\mathopen{}\left\{a^b\right\}\mathclose{} = b \cdot\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{} \]

Theorem 18 (exponential of a sum)  

The exponential of a sum is equal to the product of the exponentials of the addends:

\[\operatorname{exp}\mathopen{}\left\{a+b\right\}\mathclose{} = \operatorname{exp}\mathopen{}\left\{a\right\}\mathclose{} \cdot\operatorname{exp}\mathopen{}\left\{b\right\}\mathclose{}\]

Corollary 2 (exponential of a difference)  

The exponential of a difference is the exponential of the first term divided by the exponential of the second term:

\[\operatorname{exp}\mathopen{}\left\{a-b\right\}\mathclose{} = \frac{\operatorname{exp}\mathopen{}\left\{a\right\}\mathclose{}}{\operatorname{exp}\mathopen{}\left\{b\right\}\mathclose{}}\]

Theorem 19 (Power of a power) If \(a > 0\) and \(b, c \in \mathbb{R}\), then

\[a^{bc} = \mathopen{}\left(a^b\right)\mathclose{}^c = \mathopen{}\left(a^c\right)\mathclose{}^b\]

Example 4 (A negative base) With \(a = -1\), \(b = 2\), and \(c = \frac{1}{2}\):

\[ \begin{aligned} a^{bc} &= (-1)^{2 \cdot\frac{1}{2}} \\ &= (-1)^{1} \\ &= -1 \end{aligned} \]

but

\[ \begin{aligned} \mathopen{}\left(a^b\right)\mathclose{}^c &= \mathopen{}\left((-1)^2\right)\mathclose{}^{\frac{1}{2}} \\ &= 1^{\frac{1}{2}} \\ &= 1 \end{aligned} \]

So \(a^{bc} \neq \mathopen{}\left(a^b\right)\mathclose{}^c\) here, which is why Theorem 19 requires \(a > 0\). The third expression, \(\mathopen{}\left(a^c\right)\mathclose{}^b = \mathopen{}\left((-1)^{\frac{1}{2}}\right)\mathclose{}^2\), is not even a real number.

Corollary 3 (natural exponential of a product) \[\operatorname{exp}\mathopen{}\left\{ab\right\}\mathclose{} = (\operatorname{exp}\mathopen{}\left\{a\right\}\mathclose{})^b = (\operatorname{exp}\mathopen{}\left\{b\right\}\mathclose{})^a\]

Exercise 1 For \(b,c \in \mathbb{R}\), when does \(b^c = bc\)?

Solution 1. We only count a pair \((b, c)\) when \(b^c\) is a real number, so for \(b < 0\) we only consider integer \(c\) (R agrees: (-8)^(1/3) is NaN). With that convention, \(bc = b^c\) in each of the following cases:

1. \(c = 1\), for every \(b\).
2. \(b = 0\) and \(c > 0\), since then \(b^c = 0 = bc\). (\(b = 0\) and \(c = 0\) fails, since \(0^0 = 1 \neq 0\).)
3. \(b > 0\), \(c > 0\), \(c \neq 1\), and \(b = \operatorname{exp}\mathopen{}\left\{\frac{\operatorname{log}\mathopen{}\left\{c\right\}\mathclose{}}{c-1}\right\}\mathclose{}\). For \(b > 0\), dividing both sides of \(b^c = bc\) by \(b\) gives \(b^{c-1} = c\), which needs \(c > 0\) because \(b^{c-1} > 0\); taking logarithms then gives \((c-1)\operatorname{log}\mathopen{}\left\{b\right\}\mathclose{} = \operatorname{log}\mathopen{}\left\{c\right\}\mathclose{}\). For example, \(c = 2\) gives \(b = 2\), and indeed \(2^2 = 4 = 2 \cdot 2\).
4. \(b < 0\) and \(c\) is an odd integer with \(c \ge 3\), with \(b = -c^{1/(c-1)}\); for example, \(b = -\sqrt{3}\) and \(c = 3\) give \(b^c = -3\sqrt{3} = bc\).
5. \(b < 0\) and \(c\) is an even integer with \(c \le -2\), with \(b = -(-c)^{1/(c-1)}\); for example, \(b = -2^{-1/3}\) and \(c = -2\) give \(b^c = 2^{2/3} = bc\).

For \(b < 0\), cases 4 and 5 come from \(b^{c-1} = c\) as well: when \(c - 1\) is even, \(b^{c-1} > 0\), so \(c\) must be positive; when \(c - 1\) is odd, \(b^{c-1} < 0\), so \(c\) must be negative.

See the red contours in Figure 2 for a visualization of the \(b \ge 0\) cases.

mult_f <- function(b, c) b * c
pow_f <- function(b, c) b^c
values_b <- seq(0, 5, by = .01)
values_c <- seq(-.5, 3, by = .01)

mult_mat <- outer(values_b, values_c, mult_f)
pow_mat <- outer(values_b, values_c, pow_f)
pow_mat[is.infinite(pow_mat)] <- NA

opacity <- .3
z_min <- min(mult_mat, pow_mat, na.rm = TRUE)
z_max <- 5
plotly::plot_ly(
  x = ~values_b,
  y = ~values_c
) |>
  plotly::add_surface(
    z = ~ t(mult_mat),
    contours = list(
      z = list(
        show = TRUE,
        start = -1,
        end = 1,
        size = .1
      )
    ),
    name = "b*c",
    showscale = FALSE,
    opacity = opacity,
    colorscale = list(c(0, 1), c("green", "green"))
  ) |>
  plotly::add_surface(
    opacity = opacity,
    colorscale = list(c(0, 1), c("red", "red")),
    z = ~ t(pow_mat),
    contours = list(
      z = list(
        show = TRUE,
        start = z_min,
        end = z_max,
        size = .2
      )
    ),
    showscale = FALSE,
    name = "b^c"
  ) |>
  plotly::layout(
    scene = list(
      xaxis = list(
        # type = "log",
        title = "b"
      ),
      yaxis = list(
        # type = "log",
        title = "c"
      ),
      zaxis = list(
        # type = "log",
        range = c(z_min, z_max),
        title = "outcome"
      ),
      camera = list(eye = list(x = -1.25, y = -1.25, z = 0.5)),
      aspectratio = list(x = .9, y = .8, z = 0.7)
    )
  )

Figure 1: Graph of \(b*c\) and \(b^c\)

pow_minus_mult_f <- function(b, c) pow_f(b, c) - mult_f(b, c)

mat1 <- outer(values_b, values_c, pow_minus_mult_f)
mat1[is.infinite(mat1)] <- NA

opacity <- .3
plotly::plot_ly(
  x = ~values_b,
  y = ~values_c
) |>
  plotly::add_surface(
    z = ~ t(mat1),
    contours = list(
      z = list(
        show = TRUE,
        start = 0,
        end = 1,
        size = 1,
        color = "red"
      )
    ),
    name = "b^c - b*c",
    showscale = TRUE,
    opacity = opacity
  ) |>
  plotly::layout(
    scene = list(
      xaxis = list(
        # type = "log",
        title = "b"
      ),
      yaxis = list(
        # type = "log",
        title = "c"
      ),
      zaxis = list(
        title = "outcome"
      ),
      camera = list(eye = list(x = -1.25, y = -1.25, z = 0.5)),
      aspectratio = list(x = .9, y = .8, z = 0.7)
    )
  )

Figure 2: Graph of \(b^c - b*c\). Red contour lines show where \(b^c = b*c\).

Exercise 2 For \(a \ge 0,~b,c \in \mathbb{R}\), when does \((a^b)^c = a^{(b^c)}\)?

Solution 2. Short answer: rarely (that’s all you need to know for this course).

Long answer:

Split on whether \(a > 0\) or \(a = 0\), because the logarithm we use for \(a > 0\) is undefined at \(a = 0\).

Case \(a > 0\). By Theorem 19, \((a^b)^c = a^{bc}\), so the question becomes when \(a^{bc} = a^{(b^c)}\) (for pairs \((b, c)\) where \(b^c\) is defined). Because \(a > 0\), both sides are positive, and we can take logarithms (Theorem 17):

\[ \begin{aligned} a^{bc} &= a^{(b^c)} \\ \operatorname{log}\mathopen{}\left\{a^{bc}\right\}\mathclose{} &= \operatorname{log}\mathopen{}\left\{a^{(b^c)}\right\}\mathclose{} && \text{(take logarithms of both sides)} \\ bc \cdot \operatorname{log}\mathopen{}\left\{a\right\}\mathclose{} &= b^c\cdot \operatorname{log}\mathopen{}\left\{a\right\}\mathclose{} && \text{(logarithm of a power)} \end{aligned} \tag{1}\]

The last line of Equation 1 holds exactly when

1. \(a = 1\) (so that \(\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{} = 0\)), or
2. \(bc = b^c\) (see Exercise 1).

Case \(a = 0\). Here we cannot take logarithms, so we work from the values of powers of \(0\): \(0^s = 0\) for \(s > 0\), \(0^0 = 1\), and \(0^s\) is undefined for \(s < 0\).

• If \(b < 0\), then \(0^b\) is undefined, so \((a^b)^c\) is undefined.
• If \(b > 0\), then \((0^b)^c = 0^c\) and \(b^c > 0\), so \(0^{(b^c)} = 0\); the two sides agree exactly when \(c > 0\).
• If \(b = 0\), then \((0^0)^c = 1^c = 1\); for \(c > 0\), \(0^{(0^c)} = 0^0 = 1\), so the two sides agree, and for \(c \le 0\) they do not.

So for \(a = 0\), \((a^b)^c = a^{(b^c)}\) exactly when \(b \ge 0\) and \(c > 0\).

In particular, when \(a = 0\), \(b \ge 0\), and \(c = 0\), the two sides differ:

\[ \begin{aligned} (a^b)^c &= (0^b)^0 \\ &= 1 \end{aligned} \]

\[ \begin{aligned} a^{(b^c)} &= 0^{(b^0)} \\ &= 0^1 \\ &= 0 \end{aligned} \]

Rudin, Walter. 1976. Principles of Mathematical Analysis. 3rd ed. International Series in Pure and Applied Mathematics. McGraw-Hill.