Understanding the Number 1: Definition, Properties, Examples and Simple Exercises

Clear, student-friendly explanation of the number 1: what it is, why it is the multiplicative identity, why it is not prime, its role in counting and place value, plus examples and short exercises.

PDF

The Number 1 — Step-by-Step Explanation

Definition. The number 1 is the integer that comes after 0 and before 2. It is the first positive integer and the first natural number when you start counting from 1.

Key properties

  • Multiplicative identity. For any number a, a * 1 = a and 1 * a = a. That is, multiplying by 1 leaves a number unchanged.
  • Not prime and not composite. A prime number has exactly two distinct positive divisors (1 and itself). The number 1 has exactly one positive divisor (1), so by modern definition it is neither prime nor composite.
  • Unit in arithmetic and algebra. In ring theory the word unit means an element with a multiplicative inverse. In the integers 1 is a unit (its inverse is 1 in the rationals), and more generally 1 acts as the neutral multiplicative element in most number systems.
  • Powers and factorial. For any exponent n, 1^n = 1. Also 1! (one factorial) = 1.
  • Place value. In our base-10 place-value system, the ones place represents how many single units there are. For example in 34, the digit 4 is the ones place (4 * 1).

Simple examples

  • Counting: 1 apple means a single apple.
  • Addition and subtraction: 5 + 1 = 6; 5 - 1 = 4.
  • Multiplication and division: 7 * 1 = 7; 7 / 1 = 7.
  • Exponents: 2^1 = 2, and 1^100 = 1.

Common misconceptions

  • Some people think 1 is a prime number. It is not; primes have two distinct positive divisors.
  • 1 is the multiplicative identity, but the additive identity is 0 (because a + 0 = a).

Short exercises (with answers)

  1. Is 1 a prime number? Answer: No.
  2. Compute 1 + 1 * 3. Answer: 4 (multiply first: 1 * 3 = 3, then add 1).
  3. What is 1!? Answer: 1.
  4. If you multiply any number by 1, does it change? Answer: No.

If you want, I can explain more advanced roles of 1 (for example, its role in group and ring theory), show visual examples for young learners, or provide more practice problems. Which would you like next?

Ask a followup question