Fundamental identities

On page 52 of the textbook, you'll find a list of fundamental identities or properties of real numbers. These are statements, written using the language of algebra, that describe something which is true for all real numbers. For example, you probably notice that the following facts fit a general pattern:

2 + 3 = 3 + 2 (because both are 5);
3 + 5 = 5 + 3 (because both are 8);
4 + (−9) = (−9) + 4 (because both are −5).

This pattern continues; you can take any two real numbers, add them together in either order, and the result will be the same no matter which order you choose. If you were to list all instances of this fact, you'd never finish, because you can never list all of the expressions for real numbers. However, this fact can be summarised very simply using the language of algebra as a + b = b + a. More precisely, I should say that a + b = b + a for any real numbers a and b. Then you know that you can put in any real numbers whatsoever for a and b and the statement will be true, although for some other type of numbers you don't know if it's true. That is the first of the identities listed in the table below.

Name of the identity	Statement	For what numbers?
Commutative law of addition;	a + b = b + a;	any real numbers a and b.
Associative law of addition;	(a + b) + c = a + (b + c);	any real numbers a, b, and c.
Commutative law of multiplication;	a · b = b · a;	any real numbers a and b.
Associative law of multiplication;	(a · b) · c = a · (b · c);	any real numbers a, b, and c.
Additive identity (right);	a + 0 = a;	any real number a.
Additive identity (left);	0 + a = a;	any real number a.
Multiplicative identity (right);	a · 1 = a;	any real number a.
Multiplicative identity (left);	1 · a = a;	any real number a.
Zero law (right);	a · 0 = 0;	any real number a.
Zero law (left);	0 · a = 0;	any real number a.
Distributive law (left);	a · (b + c) = (a · b) + (a · c);	any real numbers a, b, and c.
Distributive law (right);	(a + b) · c = (a · c) + (b · c);	any real numbers a, b, and c.
Additive inverse (right);	a + (−a) = 0;	any real number a.
Additive inverse (left);	(−a) + a = 0;	any real number a.
Multiplicative inverse (right);	a · (1/a) = 1;	any nonzero real number a.
Multiplicative inverse (left);	(1/a) · a = 1;	any nonzero real number a.

(I have put in a few extra rules besides those listed in the book, but I think that these are all pretty fundamental.)

You don't really want to try to memorise this list. In any case, these are not all of the identities! (They are just in a way the most fundamental.) Instead, you should try to make these laws pare of your intuition for mathematics. Then the only thing worth memorising are the names —but I only included the names for reference, and I will not test you on them in any way.

So, the point of the commutative laws is that you can add or multiply two numbers in either order. In the associative laws, I included parentheses to indicate which addition or multiplication is to be performed first; but really the whole point of the associative laws is that it doesn't matter which you do first! So if the associative laws seem silly to you, because you don't need the parentheses in these expressions in the first place, then that's good —when a rule seems obvious, that's a sign that you understand it! The distributive law is probably the most important rules for algebra; it's complicated and may not seem very obvious at all, and yet it's used a great deal. You should think about the distributive law until it too is second nature! Notice that inverse law is true only for a nonzero real number a; if a were 0, then the statement would not make sense, because 1/0 is undefined.

Laws used in arithmetic

The ordinary rules of arithmetic that you've learned since grade school are ultimately built on identities like those above. Here are a few more identities, which follow from the fundamental ones above, but which are directly useful in arithmetic:

Statement	For what numbers?
a − b = a + (−b);	any real numbers a and b.
a/b = (1/b) · a;	any real number a and any nonzero real number b.
−(−a) = a;	any real number a.
a − b = −(b − a);	any real numbers a and b.
−a − b = −(a + b);	any real numbers a and b.
−a + b = b − a;	any real numbers a and b.
a − (−b) = a + b;	any real numbers a and b.
−a = (−1) · a;	any real number a.
a · (−b) = −(a · b);	any real numbers a and b.
(−a) · b = −(a · b);	any real numbers a and b.
(−a) · (−b) = a · b;	any real numbers a and b.
a/(−b) = −(a/b);	any real number a and any nonzero real number b.
(−a)/b = −(a/b);	any real number a and any nonzero real number b.
(−a)/(−b) = a/b;	any real number a and any nonzero real number b.
1/(1/a) = a;	any nonzero real number a.
(a · b)/b = a;	any real number a and any nonzero real number b.
(a · c)/(b · c) = a/b;	any real number a and any nonzero real numbers b and c.
(a/c) + (b/c) = (a + b)/c;	any real numbers a and b and any nonzero real number c.
(a/c) − (b/c) = (a − b)/c;	any real numbers a and b and any nonzero real number c.
(a/c) · (b/d) = (a · b)/(c · d);	any real numbers a and b and any nonzero real numbers c and d.
1/(a/b) = b/a;	any nonzero real numbers a and b.
(a/c)/(b/d) = (a · d)/(c · b);	any real number a and any nonzero real numbers b, c and d.

The first two of these are basically the definitions of subtraction and division. The next few are used to add and subtract signed numbers (the subject of Section 1.2 of the textbook); the main ideas here are that each minus sign sticks with the term to its right and two minus signs in a single term cancel each other. Then the next few are used to multiply and divide signed numbers (the subject of Section 1.3 of the textbook); the main idea here are that you can treat minus signs forming opposites as multiplication by −1 and −1 · −1 is 1. The last few are used to do arithmetic with fractions (the subject of Section 1.4 of the textbook); these are all based on the idea that division is multiplication by a reciprocal, then using the distributive law or the commutative and associative laws of multiplication.

Order of operations

There are several ways to indicate which operations should be done first. The most fundamental is to use grouping symbols, like the parentheses that I've used above. Since a lot of parentheses in a row can be confusing, there are some other grouping symbols used; here are the common ones:

round parentheses: ‘(’ and ‘)’;
square brackets: ‘[’ and ‘]’;
curly braces: ‘{’ and ‘}’.

Sometimes grouping is implicit in the position of the symbols, because there's just no other way to intepret things that makes sense:

inside absolute values;
above or below a horizontal division bar;
inside an exponent.

In principle, one should always use some grouping symbols, which I've done in the tables above. But in practice, we don't need them all the time. For example, the associative laws mean that we don't need to use grouping symbols between addition and multiplication, because it simply doesn't matter what we do first. Between addition and mutliplication, the order does matter, but we can always use the distributive law to turn an addition-first expression into a multiplication-first one. By default we always do multiplication first; then we only need grouping symbols if we (unusually) want addition first.

In general, here is the standard order of operations:

Exponentiation;
Multiplication;
Division;
Addition, subtraction, and taking opposites.

But this order can always be overridden using grouping.

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