identity element
Some numbers do nothing when you combine with them. Add 0 to any number and the number is untouched; multiply any number by 1 and it stays the same. A number that leaves every other number unchanged under a given operation is called the identity element for that operation.
For addition, the identity is 0, because a + 0 = a for every a. For multiplication, the identity is 1, because a × 1 = a for every a. Each operation has its own identity, and there is exactly one for each. The identity is the “do-nothing” anchor that makes a number system tidy and predictable.
Identity elements are quietly powerful in algebra. Writing a number times 1, or adding a clever 0, lets you change a fraction's form without changing its value or add and subtract the same quantity to set up a new manipulation. The very idea of an inverse depends on the identity: an inverse is what you combine with something to get back to the identity.
To rewrite 3/5 with denominator 20, multiply by the multiplicative identity in disguise: 3/5 × (4/4) = 12/20. Since 4/4 = 1, the value is unchanged.
Multiplying by a disguised 1 changes a fraction's appearance but not its value.
The additive identity 0 and the multiplicative identity 1 are different numbers, which is why 0 ≠ 1 is built into the rules of arithmetic. This same idea of an identity reappears later as the identity of a group and the identity matrix.