Maths is a subject that relies heavily on logic and hence it becomes very important to define terms in math.
Why are definitions important?
Why is it essential to define a term very strongly? Why can’t the understanding of something be enough?
Is 1 a prime number?
No, because it is divisible only by 1 and no other number. So, 1 is not a prime number.
But one can argue that -1 X -1 = 1 and so there are two factors of 1, -1 and 1. Also, 2 X 0.5 = 1, 4 X 0.25 = 1 and this way there can be more than 1 factors+ of 1. And so, 1 can be called a composite number.
But still, 1 is not a composite number. Why? The answer lies in the definition of a prime number.
A prime number is defined as “A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself.”
+ Now one can even argue that 0.5 cannot be called a factor of 1 because a factor is defined as “A factor is a portion of a quantity, usually an integer or polynomial that, when multiplied by other factors, gives the entire quantity.” However, according to this definition, 0.5 can be called a factor of 1. But, in number theoretic usage, a factor of a number n is equivalent to a divisor of n (Ore 1988, p.29; Burton 1989, p. 26). This brings us back to one more term and its definition, divisor. A divisor of a number n is a number d which divides n, also called a factor. This way, we can go round and round the logical bush. A point will remain valid in math until no one else finds a logical way of proving it wrong.
Why are definitions important?
Why is it essential to define a term very strongly? Why can’t the understanding of something be enough?
Is 1 a prime number?
No, because it is divisible only by 1 and no other number. So, 1 is not a prime number.
But one can argue that -1 X -1 = 1 and so there are two factors of 1, -1 and 1. Also, 2 X 0.5 = 1, 4 X 0.25 = 1 and this way there can be more than 1 factors+ of 1. And so, 1 can be called a composite number.
But still, 1 is not a composite number. Why? The answer lies in the definition of a prime number.
A prime number is defined as “A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself.”
+ Now one can even argue that 0.5 cannot be called a factor of 1 because a factor is defined as “A factor is a portion of a quantity, usually an integer or polynomial that, when multiplied by other factors, gives the entire quantity.” However, according to this definition, 0.5 can be called a factor of 1. But, in number theoretic usage, a factor of a number n is equivalent to a divisor of n (Ore 1988, p.29; Burton 1989, p. 26). This brings us back to one more term and its definition, divisor. A divisor of a number n is a number d which divides n, also called a factor. This way, we can go round and round the logical bush. A point will remain valid in math until no one else finds a logical way of proving it wrong.
