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.
1 comment:
Interesting post..so it seems to me that definitions are important because they make communication possible...or quality dialogue within a discipline possible. And the term definition can be extended to include words, jargon etc. You and I can discuss a problem or an issue or share our thoughts only if we agree on certain symbols that have "common meanings". Now if we were just interested in "killing some time" and moving on, it wouldnt matter so much that rigorous definitions didnt exist. But if we were really serious and wanted to take this somewhere, we would need to start agreeing on some definitions. What puzzles me however is...Is maths real? I wonder how complex numbers got "invented" for example. The sad thing about maths and science education is that it is taught without any context to the history of the discipline, so a lot of all this seems weird to me.
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