Useful Unicode Symbols:
logic: ∨ ∧ ¬ ∀ ∃ → ↔ ≡ ⊕↦
arithmetic: ÷ · × √ ∛ ∜ ± ⌈⌉ ⌊⌋↑ ⋆ ∘ ⌊⌋ ⌈ ⌉
comparison: ≠ ≤ ≥ ≃ ≅
simple fractions: ½ ⅓ ⅔ ¼
sets and set related: ∅ ℕ ℤ ℚ ℝ ℂ ∈ ∉ ∩ ∪ ⊂ ⊃ ⊆ ⊇
subscripts: ₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋ ₌ ₍ ₎ ₐ ₑ ₒ ₓ ₕ ₖ ₗ ₘ ₙ ₚ ₛ ₜ
superscripts: ⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁼ ⁽ ⁾ ⁿ ⁱ ˣ ᵃ ᵇ ᶜ ᵈ ᵉ ᶠ ᵍ ʰⁱʲ ᵏ ˡ ᵐ ⁿ ᵒ ᵖ ʳ ˢ ᵘ ᵛ ʷ ˣ ʸ ᶻ
lowercase greek letters: α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς τ υ φ χ ψ ω
uppercase greek letters: Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω
calculus: ∫ ∞
geometry: □ △ ○
A List of Bad Math Habits to Unlearn
You may come into this course with some "bad math habits" or incorrect ideas from previous math courses. Please make an effort to un-learn such bad math habits.
Use equal signs between equal quantities, not arrows
It two quantities are equal, use the equal sign "=" to indicate this. Do not write any other symbol.
Some math students like to write right arrows between equal quantities. The right arrow suggests the informal meaning of "and the next step is". If you do this, it is a habit you must unlearn, because the right arrow has a different canonical meaning.
Do not write "equations" like x² + 5x + 6 → (x+2)(x+3). Use equal signs instead: x² + 5x + 6 = (x+2)(x+3).
We learn in MAT 243 that the right arrow is formally a logical operator, i.e. it connects two propositions and forms a new formal proposition. This will be explained in the lectures.
There is also an informal use of the right arrow which is not supposed to denote creation of a new formal statement, but expresses a logical consequence of an established fact in a chain of deductions. When the right arrow is used in this informal capacity, the double right arrow is usually preferred. For example,
(x+2)(x+3) = 0 ⇒ x = -2 or x = -3.
This says: if we know that the product of (x+2) and (x+3) is zero, then x must be -2 or x must be -3.
Do not use the equal sign to denote equivalence of algebraic equations or inequalities
The equations 2x = 4 and x = 2 are algebraically equivalent. They have the same solution. Generally, two equations are called equivalent if they have the same set of solutions.
However, that does not mean that the sides in equivalent equations are all equal: 2x = 4 = x = 2 is incorrect. x cannot be be simultaneously equal to 2 and to 4. The notation (2x = 4) = (x = 2) is just as bad. It asserts that two equations are equal, but what does it mean for two equations to be equal? Equal as strings objects? The strings "2x = 4" and "x=2" are not equal.
It is common to use the double bidirectional arrow ⇔ to denote algebraic equivalence. To say that 2x = 4 and x = 2 are equivalent, write
2x = 4 ⇔ x = 2.
Everything just said also applies to inequalities. For example, 2x < 10 is equivalent to x < 5. However, you must not write 2x < 10 = x < 5.
That string of equalities and inequalities says: 2x < 10 and 10 = x and x < 5. Clearly, neither inequality implies that x is 10.
To express the meaning that 2x < 10 is equivalent to x < 5, write 2x < 10 ⇔ x < 5.
Use parentheses properly, and do not use redundant parentheses. Do not use parentheses as the multiplication operator.
You may have learned in lower-level math courses that parentheses indicate multiplication. This is incorrect. Parentheses override the default order of operations. For example, if you wish to add the numbers 1 and 2, and then raise the sum to the power of x, you need to write (1+2)ˣ. Exponentiation has a higher order of operations than addition, so you need to put the sum in parentheses.
If you have learned the bad habit of using parentheses as standard packaging for factors in a product, or generally placing unnecessary parentheses, please make an effort to un-learn this. You should write the product of 1 and 2 as 1·2, not as 1(2) or (1)(2). The latter two notations are not technically wrong, but they suggest an incorrect understanding of the role parentheses play in algebraic notations. In addition, redundant parentheses make algebraic expressions harder to read (and take longer to write). For example, 1 + 2x is easier to read than ((1) + (2)(x)), and quicker to write as well, even though the latter is technically correct.
In mathematical writing, use correct singular and plural forms
In mathematics, we typically speak of parentheses. This is a plural form. () are parentheses. The plural form is more common because in correct algebraic expression, each opening parenthesis "(" is matched by a closing parenthesis ")". Parenthesis is the singular form.
If you learned that the singular form is "parenthesee", please unlearn this.
Maxima, minima and extrema are all plural forms. The corresponding singular forms are maximum, minimum and extremum. Example:
"The absolute minimum of f(x) = x² is at x = 0."
Avoid common errors in quadratic equation solving
Quadratic equations often have two solutions. More specifically, x² = a, for a positive real number a, always has two solutions, one positive, one negative. Do not overlook the negative solution. For example, x² = 9 has the solutions x = 3 and x = -3.
The square root symbol √ is not multi-valued. The square root of 9 is the number 3. It does not mean ±3.
If a quadratic equation has a linear term, then you need to complete a square or use the quadratic equation. You cannot solve such an equation by simply factoring out the independent variable from the quadratic and linear term.
For example, consider x² + 2x = 15. Factoring out an x on the left side leads to the dead end x(x+2) = 15.
To solve the equation x² + 2x = 15, complete the square and write the equation equivalently as x² + 2x + 1 = 16. Then factor the left side as (x+1)². This leads to: x + 1 = 4 or x + 1 = -4, i.e. x = 3 or x = -5.
Seemingly alternatively, you can use the quadratic formula. However, the quadratic formula is nothing but the solution via completing the square, compressed into one formula.
Do not confuse multiplication and exponentiation
2ⁿ does not mean 2n. The superscript indicates exponentiation, which, for positive integer arguments, is repeated multiplication of the base by itself.
For example, if we take n = 10, then
2ⁿ = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 = 1024 while 2n = 2 · 10 = 20.
Do not confuse the words principle and principal.
Both of these words appear in lower-level university mathematics and must be carefully distinguished.
A principle is a fundamental rule. For example, in this course, we learn the multiplication principle, and the principle of induction.
Principal is an adjective that means first, foremost, most important. It can also be used as a noun, and then refers to someone who holds a leading position, like the school principal.
Principal appears in mathematics as well. There are a number of situations where a mathematical problem has more than one solution, but one is considered preferred by mathematicians, so they call that preferred solution the principal solution or principal value.
Avoid intermediate rounding in calculations
Intermediate rounding means that you round a number obtained from a calculation and then perform more calculations with that rounded number. Rounding errors can compound, and sometimes be greatly amplified by subsequent calculations. Rounding the input to a function to an inch does not mean that the output will be accurate to an inch. It could be off by the radius of the observable universe.
The derivative of a function can help us understand this phenomenon. One way to think about the derivative f'(a) is that it is the factor by which it amplifies very small rounding errors in the input a. If f'(a) is 1 million, it means that a rounding error in the input a of 0.001 will cause the output to be off by 1000.
Always calculate with exact numbers, or at least with the best approximation your calculator will give you. Here is an example:
x is 1/81921 and y = 2/x. Working with exact numbers, we get the correct value y = 2 · 81921 = 163842.
However if you work out a calculator approximation of x, and round it like this: x = 0.0000122, then 2/x turns out roughly 163840.419. Compare this to the correct answer 163842.
Intermediate rounding is a likely cause of a common online homework "mystery" in math classes. The system accepted your first answer to a 2-part problem, but not your second answer, even though you can find no mistake in your algebra.
When making substitutions in integrals, do not use the misleading terminology "u-substitution".
Some calculus teachers now teach the substitution rule for integration as "u-substitution". This is misleading terminology. How you want to name a new variable in an integral is entirely up to you; you can use any letter you wish. The validity of a mathematical method does not depend on a variable name. If an integral is already written in terms of a variable u, then you would have to use a different letter than u.
A new variable may also have a specific meaning in the context of the application, in which case you may want to use a letter associated with that meaning, not necessarily "u".
It is best if you speak of using the "substitution rule" or simply "substitution".