Algebraic notations

This is mostly a repetition.

The basic operations

Let's start with the four basic algebraic operations:

Addition a+bsummand plus summand equals sumSubtraction a−bminuend minus subtrahend equals differenceMultiplication a⋅bfactor times factor equals productDivision a:bdividend divided by divisor equals quotient\begin{array}{lll} \text{Addition } a+b & \text{summand plus summand equals sum} \\ \text{Subtraction } a-b & \text{minuend minus subtrahend equals difference} \\ \text{Multiplication } a\cdot b & \text{factor times factor equals product}\\ \text{Division } a:b & \text{dividend divided by divisor equals quotient} \end{array}

Not so important are the terms minuend, subtrahend, dividend and divisor. However, you should know the meaning of the terms summand, factor, product, difference, sum and quotient.

Equal terms

We often say that one term is equal to some other term, or one term is equivalent to some other term or some term is the same as some other term. What we mean by that is best shown with an example. The term

a+b−3a+2b−aa+a+1a+b-3a+2b-\frac{a}{a}+a+1

is indeed equal to the term

3b−a3b-a

We write this as follows:

a+b−3a+2b−aa+a+1=3b−aa+b-3a+2b-\frac{a}{a}+a+1 = 3b-a

What we mean by this is that for every number we use for aa and bb, the two terms must have the same value. For example, if we put a=5a=5 and b=6b=6, we get for the left side

5+6−3⋅5+2⋅6−55+5+1=135+6-3\cdot 5+2\cdot 6-\frac{5}{5}+5+1=13

and for the right side we get also

3⋅6−5=133\cdot 6-5=13

This has to be true for every pair of numbers we use for aa and bb.

Example 1

Show that x+2y≠2x+yx+2y \neq 2x+y.

Hint: find two numbers such that the left side differs from the right side.

Notations involving multiplication

It is really important for you to know the different ways of how we can write terms. There a few rules or conventions:

  1. Multiplication with or without dot: Since variables always consist of one letter, we take the "word" abab to be the product of the variables aa and bb:

    ab=a⋅b\boxed{ab=a\cdot b}

    and similar

    abc=a⋅b⋅cabc=a\cdot b\cdot c

    and so on. This is also true when numbers are involved:

    3ab=3⋅a⋅b3ab=3\cdot a\cdot b

    and

    2a3b=2⋅a⋅3⋅b2a3b = 2\cdot a\cdot 3\cdot b

    Note that we try to avoid writing a number after the variable without a dot (e.g. a3a3 for aa times 33), and would either write a⋅3a\cdot 3 or even better 3a3a.

  2. Consecutive brackets are also meant to be multiplied:

    (−2)(a+b)=(−2)⋅(a+b)(-2)(a+b)=(-2)\cdot (a+b) (a+b)(a−b)=(a+b)⋅(a−b)(a+b)(a-b)=(a+b)\cdot (a-b)

    and the same is true for a variable or number followed by a bracket:

    3(a+b)=3⋅(a+b)3(a+b)=3\cdot (a+b) a(a+b)=a⋅(a+b)a(a+b)=a\cdot (a+b)

  3. Multiplication with a negative number: The same is true for negative numbers, where we often form a parenthesis around the negative number to indicate that the minus sign belongs to the number

    −2ab=−2⋅a⋅b=(−2)⋅a⋅b-2ab=-2\cdot a\cdot b = (-2)\cdot a\cdot b

    If a negative number occurs in the middle of a multiplication, then it is necessary to use parentheses:

    a(−2)b=a⋅−2⋅ba(-2)b=a\cdot -2\cdot b

    We need the parentheses because

    a−2b=a−2⋅ba-2b=a-2\cdot b

    which is a subtraction and is not the same term as a(−2)ba(-2)b.

  4. Multiplication by 11 and −1-1: A special case is the factor 11 and −1-1, since the "1" is often omitted

    1a=a\boxed{1a=a}

    and

    −1a=−a\boxed{-1a=-a}

    The holds for brackets, so

    1(a+b)=(a+b)1(a+b)=(a+b)

    and

    −1(a+b)=−(a+b)-1(a+b)=-(a+b)