Laws of Multiplication

Laws similar to those for addition also apply to multiplication. Special attention should be given to the multiplicative identity and inverse, M3 and M4.

M1. The product of any two real numbers a and b is again a real number, denoted a·b or ab.

M2. No matter how terms are grouped in carrying out multiplications, the product will always be the same: (ab)c = a(bc). This is called the associative law of multiplication.

M3. Given any real number a, there is a number one (1) called the multiplicative identity, such that a(1) = 1(a) = a.

M4. Given any nonzero real number a, there is a number (a^-1), or (1/a), called the multiplicative inverse, such that a(a^-1) = (a^-1)a = 1.

M5. No matter in what order multiplication is carried out, the product will always be the same: ab = ba. This is called the commutative law of multiplication.

Any set of elements obeying these five laws is said to be an Abelian, or commutative, group under multiplication. The set of all real numbers, excluding zero—because division by zero is inadmissible—forms such a commutative group under multiplication.