In mathematics, a multiplicative identity is a special number that, when multiplied by any other number, leaves the other number unchanged. **The multiplicative identity is always the number $1$.** Here's why: - **Definition of Identity:** An identity element is a special element in a set that, when combined with any other element in the set using a particular operation, leaves the other element unchanged. - **Multiplication:** For any real number '$a: - $a * 1 = a$ - $1 * a = a$ **Examples** - $7 * 1 = 7$ - $3.14159 * 1 = 3.14159$ - $-15 * 1 = -15$ **Key Properties** - **Uniqueness:** There's only one multiplicative identity within a set of numbers (like the real numbers or complex numbers). - **Role in Equations:** The multiplicative identity is essential for solving equations. For example, to isolate '$x in the equation $5x = 35$, we divide both sides by $5$, which is essentially multiplying both sides by the multiplicative inverse of $5 (1/5)$. **Other Identities** - **Additive Identity:** The [[additive identity]] is $0$ (zero), since adding zero to any number leaves it unchanged. - **Matrix Identity:** For matrices, the [[Identity matrix]] is a specific matrix (with 1s on the diagonal and 0s elsewhere) that behaves similarly to the number 1 for matrices. - # References ```dataview Table title as Title, authors as Authors where contains(subject, "multiplicative") or contains(subject, "identity") sort modified desc, authors, title ```