1.2 General Terminology: Terminology in Context

Mathematical communication uses verbs in specific, recurring ways. Unlike in everyday English, each verb signals a specific communicative function; for example, whether the person is making a claim, giving a reason, or drawing a conclusion. Learning these verbs as groups will help you both understand mathematical texts and write your own.

1. Observing and noting: These verbs introduce something the reader should pay attention to. They are often used at the start of a sentence.
   • We note that the denominator must be non-zero.
   • We observe that both sides of the equation are equal.
   • We see that the function is increasing on this interval.
   • Notice that the two expressions differ only by a constant.
   • It is worth remarking that this result holds for all integers.
2. Assuming and supposing: These verbs introduce a condition or hypothesis that is taken to be true for the purposes of the argument.
   • Suppose we have two real numbers [latex]a[/latex] and [latex]b[/latex].
   • Assume that [latex]n[/latex] is a positive integer.
   • Let [latex]f[/latex] be a continuous function on the interval [latex][a, b][/latex].
   • Given a non-zero constant [latex]c[/latex]…
   • Consider the case where the two values are equal.

Note: Suppose, assume, and let are near-synonyms in mathematical writing, but suppose is more common when introducing a scenario, assume when stating a simplifying condition, and let when naming a variable or object.

3. Defining and introducing notation: These verbs are used when a term or symbol is given a precise meaning for the first time.
   • We define the derivative of [latex]f[/latex] as the limit of the difference quotient.
   • We denote the set of all real numbers by [latex]\mathbb{R}[/latex].
   • We call this value the constant of integration.
   • This quantity is known as the arithmetic mean.
   • This operation is referred to as scalar multiplication.
   • We write the result in the form [latex]ax + b = 0[/latex].

Note: The pattern for denote is always 'we denote X by Y', and the pattern for call is 'we call X the Y'. These are fixed collocations and should be learnt as chunks.

4. Claiming and asserting: These verbs introduce a mathematical statement that is being put forward as true.
   • We claim that the sequence converges to zero.
   • We assert that no such value of [latex]x[/latex] exists.
   • It follows that the sum must be finite.
   • We conclude that the two expressions are equivalent.
   • This implies that [latex]a = b[/latex].
   • This step gives us the value of the constant.
   • Solving the equation yields two distinct roots.

Note: Implies, gives, and yields are often used with a mathematical expression as the subject. For example, 'this step implies that [latex]a = b[/latex]'. They signal a logical consequence.

5. Reasoning and justifying: These verbs and phrases signal that an explanation or justification is being offered.
   • This is because the sum of two even numbers is always even.
   • This follows from the definition of a limit.
   • This result can be shown by induction.
   • This holds for all values of [latex]n[/latex] greater than zero.
   • This is a consequence of the fundamental theorem of calculus.
6. Generalising and extending: These verbs indicate that a result or method applies more broadly.
   • We can generalise this result to any real number.
   • We extend this approach to functions of two variables.
   • More generally, this method works for any continuous function.
   • We can apply the same argument to the case of complex numbers.
7. Illustrating and exemplifying: These verbs introduce examples, diagrams, or concrete cases.
   • We illustrate this with a simple example.
   • We demonstrate the method by considering a specific case.
   • We exemplify this by computing the derivative of [latex]x^2[/latex].
   • As a counter-example, consider the function [latex]f(x) = |x|[/latex], which is continuous but not differentiable at [latex]x = 0[/latex].
   • To verify, we substitute the value back into the original equation.
8. Computing and manipulating: These verbs describe concrete mathematical operations performed on expressions or objects.
   • We compute the value of the integral.
   • We evaluate the expression at [latex]x = 1[/latex].
   • We substitute [latex]x = 0[/latex] into the equation.
   • We rearrange the inequality to isolate [latex]x[/latex].
   • We solve the equation for the unknown.
   • We simplify the expression by cancelling common factors.
   • We expand the brackets.
   • We factorise the quadratic.
9. Verifying and checking: These verbs signal that a result is being confirmed or tested.
   • We verify that the answer satisfies the original equation.
   • We check that the conditions of the theorem are met.
   • We confirm that the two sides of the identity are equal.
   • We test whether the series converges.
10. Concluding and summarising: These verbs close an argument or summarise a result.
    • We conclude that the function has exactly one root.
    • It follows that the series diverges.
    • We have shown that the result holds for all positive integers.
    • This completes the proof.
    • In summary, a function is continuous if it has no breaks or jumps.