Algebraic Properties of R On the set R real numbers there are two binary operations, denoted by +
and • and called addition and multiplication, respectively. These operations satisfy the following
properties:
i. a + b =b + a for all a, b in R (commutative property of addition);
ii. (a + b) + c = a + (b + c) for all a, b, c in R (associative property of addition);
iii. there exists an element 0 in R such that 0 + a = a and a + 0 = a for all a in R (existence of a zero
element);
iv. for each a in R there exists an element -a in R such that a + (-a) = 0 and (-a) + a = 0 (existence of
negative elements);
v. a • b =b • a for all a, b in R (commutative property of multiplication);
vi. (a • b) • c = a • (b • c) for all a, b, c in R (associative property of multiplication);
vii. there exists an element 1 in R distinct from 0 such that 1 • a = a and a • 1= a for all a in R
(existence of a unit element);
viii. for each a ≠ 0 in R there exists an element 1/a in R such that a • (1/a) = 1 and
(1/a) • a = 1 (existence of reciprocals);
ix. a • (b + c) = (a • b) + (a • c) and (b + c) • a = (b• a) + (c • a) for all a, b, c in R (distributive
property of multiplication over addition).
1. a + b ϵ R, a • b ϵ R. (Closure)
This means we can add and multiply real numbers. We can also subtract real numbers and we
can divide as long as the denominator is not 0.
2. a + b = b + a, a · b = b · a. (commutative Law)
This means when we add or multiply real numbers, the order doesn’t matter.
3. (a + b) + c = a + (b + c), (a · b) · c = a · (b · c). (Associative Law)
We can thus write a + b + c or a · b · c without worrying that different people will get different
results.
4. a · (b + c) = a · b + a · c, (a + b) · c = a · c + b · c. (Distributive Law)
The distributive law is the one law that involves both addition and multiplication. It is used in
two basic ways: to multiply two factors where one factor has more than one term and to factor
out a common factor when we add or subtract several terms, all of which contain a common
factor
5. a + 0 = 0 + a = a, a · 1 = 1 · a = a (0 is the additive identity, 1 is the multiplicative identity.)
Additive Inverse: Every a ∈ R has an additive inverse, denoted by −a, such that a+(−a) = 0, the
additive identity.
Multiplicative Inverse: Every a ∈ R except for 0 has a multiplicative inverse, denoted by aˉ¹ or
1/a, such that a · aˉ¹ = aˉ¹ · a = 1, the multiplicative inverse.
6. Cancellation Law for Addition: If a + c = b + c, then a = b. This follows from the existence of an
additive inverse (and the other laws), since if a + c = b + c, then a + c+ (−c) = b + c+ (−c), so a + 0 =
b + 0 and hence a = b.
7. Cancellation Law for Multiplication: If a · c = b · c and c ≠ 0, then a = b.
This follows from the existence of a multiplicative inverse for c (and the other laws), since if a · c
= b · c, then a · c · cˉ¹ = b · c · cˉ¹, so a · 1 = b · 1 and hence a = b.
We can now see why multiplication by −1 yields the additive inverse of a number: a+(−1)·a = 1 ·
1 + (−1) · a = (1 + (−1)) · a = 0 · a = 0.
We can also see why the product of a positive number and a negative number must be negative,
and the product of two negative numbers is positive. More generally, we can see that (−a) · b =
−a · b as follows: a · b + (−a) · b = (a+ (−a)) · b = 0· b = 0, so (−a) · b must be the additive inverse of
a · b, in other words, −a · b.
