Division Properties of One
Division Properties of One covers two rules for dividing with 1: any non-zero number divided by itself equals 1 (a ÷ a = 1), and any number divided by 1 equals itself (a ÷ 1 = a). From OpenStax Prealgebra 2E: 23 ÷ 23 = 1 and 35 ÷ 1 = 35. These properties appear constantly in fraction simplification — dividing numerator and denominator by the same number is equivalent to dividing by a form of 1. Mastering this property speeds up reducing fractions and understanding why equivalent fractions are equal.
Key Concepts
Property Dividing any number(except 0) by itself produces a quotient of 1. Also, any number divided by 1 produces a quotient of the number. These two ideas are stated in the Division Properties of One.
Examples Any number divided by itself is 1, so $23 \div 23 = 1$. Any number divided by one is itself, so $\frac{35}{1} = 35$. Using both properties, we know $14 \div 1 = 14$ and $14 \div 14 = 1$.
Explanation These are awesome shortcuts! If you divide a number by itself, the answer is always 1. If you divide any number by 1, the number stays the same. These rules make some division problems super fast to solve.
Common Questions
What is any number divided by itself?
Any non-zero number divided by itself equals 1. For example, 23 ÷ 23 = 1.
What is any number divided by 1?
Any number divided by 1 equals the number itself. For example, 35 ÷ 1 = 35.
Is 0 ÷ 0 equal to 1?
No. 0 ÷ 0 is indeterminate, not 1. The property a ÷ a = 1 applies only when a ≠ 0.
How do division properties of 1 relate to simplifying fractions?
When you cancel common factors in a fraction, you are dividing numerator and denominator by the same number — dividing by a fraction equal to 1.
What is the multiplicative identity and how does it relate?
The number 1 is the multiplicative identity: any number times 1 equals itself. Division by 1 is the inverse of multiplying by 1.
Give an example using both properties together.
14 ÷ 1 = 14 (divided by 1 gives itself). 14 ÷ 14 = 1 (divided by itself gives 1).