Precision

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Important points about being precise in mathematical writing.

In mathematical writing, we aim to use vocabulary, grammar, and notation happily vaguely precisely quickly , meaning with careful attention to details.

The following problems are intended to help you do and write mathematics more precisely.

In mathematics, we distinguish expressions from equations and inequalities. Equations and inequalities can be complete sentences. Expressions are noun phrases that may be used within sentences. For example:

Let $n$ be the number of nickels and $d$ be the number of dimes. Then the number of coins is $\answer {n+d}$ , and their value (in cents) is $\answer {5n+10d}$ .

(a): The statement $a+b = b+a$ is called the $\answer [format=string]{commutative}$ property of $\answer [format=string]{addition}$ .
(b): The statement $(a+b)+c = a+(b+c)$ is called the $\answer [format=string]{associative}$ property of $\answer [format=string]{addition}$ .
(c): The statement $ab = ba$ is called the $\answer [format=string]{commutative}$ property of $\answer [format=string]{multiplication}$ .
(d): The statement $(ab)c = a(bc)$ is called the $\answer [format=string]{associative}$ property of $\answer [format=string]{multiplication}$ .
These equations are called functions expressions identities relations because they are true for all values of the variables.

Notice that each of these properties involves one operation: either addition or multiplication, but not both.

Which of the following statements are identities (i.e., always true)?

$(a+b)^2=a^2+b^2$ $3(x+2)=3x+6$ $m(pq)=(mp)(mq)$ $2x+3x=(2+3)x$ $(3+x)(4+x)=12+7x+x^2$ $\sqrt {x^2+y^2}=x+y$ $(4x)^3=4^3 x^3$

Which of the following are examples of the distributive property?

$(a+b)^2=a^2+b^2$ $3(x+2)=3x+6$ $m(pq)=(mp)(mq)$ $2x+3x=(2+3)x$ $(3+x)(4+x)=12+7x+x^2$ $\sqrt {x^2+y^2}=x+y$ $(4x)^3=4^3 x^3$

The statement $a(b+c)=ab+ac$ is called the $\answer [format=string]{distributive}$ property of $\answer [format=string]{multiplication}$ over $\answer [format=string]{addition}$ .”

Correct. We often abbreviate this as “the distributive property,” but it is important to keep in mind the full name because it shows how these two operations work together.

The expression simplification $3x+4x+y+5y=7x+6y$ is an example of a procedure often called “collecting $\answer [format=string]{like terms}$ ”, but in fact is it yet another example of the $\answer [format=string]{distributive}$ property.

Correct. Here are some details: $3x+4x+y+5y=(3+4)\answer {x}+\left (\answer {1}+5\right )y=7x+6y$

A rational number is any number that can be expressed as $\answer {\frac {a}{b}}$ where $a$ and $b$ are $\answer [format=string]{integers}$ and $b$ $=$ $<$ $>$ $\ne$ $\answer {0}$ .

A function is a rule that assigns to each number input output formula exactly one number input output formula . The set of all input values is called the $\answer [format=string]{domain}$ of the function. The set of all output values is called the $\answer [format=string]{range}$ of the function.

Consider the equation $A+b+2=2a+B+5$ . Solve the equation for $A$ .

$A=\answer {2a+B-b+3}$ .

Are upper- and lower-case letters interchangeable?

Yes. Always distinguish between upper- and lower-case letters. Assume that $A\ne a$ , $B\ne b$ , etc. Do not use them interchangeably.

Sandy is accepting orders for boxes of cookies, which are $\$4$ each plus $\$6$ for shipping. As an example, he writes the calculation for an order of $5$ boxes as follows: $5\cdot 4 = 20 + 6 = 26.$ Has Sandy written this in a way that is mathematically precise?

Yes No It depends

Sandy has misused the equals sign by writing $5\cdot 4 = \answer {20 + 6}$ , which is false.

Here are two possible ways to fix this:

One-sentence method: $5\cdot 4 + 6 = 20 + 6 = 26$ .
Two-sentence method: $5\cdot 4 = 20$ . Then $20 + 6 = 26$ .

In this class, we avoid “cross multiplication,” because the procedure is too often misused, and few students have any idea why it works. Instead, we apply a general method for solving equations involving fractions: $\answer [format=string]{multiply}$ the equation by the $\answer [format=string]{denominators}$ , either one at a time or all at once (if you are brave).

Is $1$ prime, composite, neither, or both?

prime composite neither both

Correct! We want the prime factorization of a number to be unique, up to the order of the factors. Clearly $1$ is not composite. If $1$ were prime, then the prime factorization of $12$ could be $2\cdot 2\cdot 3 \cdot 1\cdot 1$ , or $2\cdot 2\cdot 3$ times as many $1$ ’s as you like. Better to decide that $1$ is not prime.

With this decision, what can be said about the typical definition: “A counting number is prime if its only factors are $1$ and itself.”

It is correct. It is ambiguous and therefore not satisfactory. It is incorrect.

That’s right. A statement is ambiguous if it can be interpreted in more than one way. Definitions should be clear and unambiguous.

Here is a better definition: “A counting number is prime if it has $\answer {2}$ factors.”

In math class, you might hear the following definition: $a^n\text { means }a\text { multiplied by itself }n\text { times.}$ Is the definition true or false?

True False

Correct! For example, in $3^5=3\cdot 3\cdot 3\cdot 3\cdot 3$ , there are only $\answer {4}$ multiplications.

Furthermore, when the exponent is negative or a fraction, it is hard to think about repeated multiplication.

So we need a better definition. Here is one possibility:

When $n$ is a , $a^n$ means the $\answer [format=string]{product}$ of $\answer {n}$ (copies of) $\answer {a}$ .