Chang's Math blog

Monday, October 5, 2026

P5 Geometry

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With same Perimeter, which has the biggest Area?

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With same Area, which has the longest Perimeter?

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Does Prime Numbers apply to real numbers?

No, the concept of prime numbers does not apply to all real numbers. It applies strictly to integers (specifically, whole numbers greater than 1).

Why Primes Only Apply to Integers

By definition, a prime number is a whole number that has exactly two distinct positive divisors: 1 and itself.

If you try to apply the concept of "divisibility" to real numbers (like decimals or fractions), the definition breaks down completely:

Any non-zero real number can be divided perfectly by any other non-zero real number.

For example, \(\mathbf{7}\) is prime because its only integer factors are 1 and 7. But in real numbers, you can divide 7 by 2 to get exactly 3.5, or divide it by 3 to get \(2.\bar{3}\).

Because everything is infinitely divisible in the real number system, the idea of a "factor" loses its meaning.

Abstract Algebra Exception

In advanced mathematics (abstract algebra), mathematicians generalize the concept of primes to other systems using integers. For example, Gaussian integers (complex numbers where both the real and imaginary parts are integers, like \(a + bi\)) have their own unique set of "Gaussian primes." However, this still relies on integer-like structures, not continuous real numbers.

Sunday, October 4, 2026

is 997 a prime number?

Yes, 997 is a prime number. [1]

Key Details

Factors: It is divisible only by 1 and 997.

Position: It is the 168th prime number overall and the largest 3-digit prime number.

Square Root Check: The square root of 997 is about 31.58. Testing prime numbers up to 31 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31) shows no whole-number divisors

https://www.google.com/search?q=is+997+a+prime+number&ie=UTF-8&oe=UTF-8&hl=en-sg&client=safari&iga=1&utm_campaign=safari_share_1

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what is the best way to test.if a number is prime?

The best way to test if a number is prime depends entirely on the size of the number you are testing. For everyday programming and small numbers, trial division up to the square root is the most efficient and definitive method. For massive numbers used in cryptography, advanced probabilistic algorithms like the Miller-Rabin primality test are preferred.

Summary of Primality Tests

Method	Best Used For	Type	How It Works
Trial Division (\(\sqrt{n}\)) [1]	Small numbers (under \(10^{12}\)) [1]	Deterministic (100% exact)	Checks if \(n\) is divisible by any integer from 2 up to \(\sqrt{n}\).
Miller-Rabin [1]	Large numbers (cryptography) [4]	Probabilistic (99.99%+ accurate)	Uses modular exponentiation properties to rapidly filter out composites.
AKS Test [1]	Theoretical computer science	Deterministic (100% exact)	First general, polynomial-time exact test, but too slow for practical use.

1. The Practical Standard: Optimized Trial Division

To test a smaller number \(n\) manually or in code, you do not need to check every number up to \(n\). You only need to check up to \(\sqrt{n}\) because if a number has a factor larger than its square root, it must also have a matching factor smaller than its square root.

You can optimize this further by checking 2 and 3 first, and then skipping all other even numbers and multiples of 3. Every prime number greater than 3 can be written in the form \(6k \pm 1\).

Step-by-Step Logic:

If \(n \le 1\), it is not prime.

If \(n \le 3\), it is prime (if 2 or 3).

If \(n\) is divisible by 2 or 3, it is not prime.

Check divisibility by numbers of the form \(5, 7, 11, 13, 17, 19...\) (stepping by 6 each time) up until the test number squared is greater than \(n\).

Optimized Python Code Example:

python

def is_prime(n):
    if n <= 1:
        return False
    if n <= 3:
        return True
    if n % 2 == 0 or n % 3 == 0:
        return False
    
    # Check 6k +/- 1 up to sqrt(n)
    i = 5
    while i * i <= n:
        if n % i == 0 or n % (i + 2) == 0:
            return False
        i += 6
    return True

Use code with caution.

2. The Cryptographic Standard: Miller-Rabin Test

When dealing with numbers that are hundreds of digits long (like RSA encryption keys), trial division takes too long. Computer scientists use the Miller-Rabin test.

It is a probabilistic test, meaning it can tell you with 100% certainty if a number is composite, but only with high probability if it is prime.

By running the test for multiple "rounds" using different random bases, the chance of a composite number sneaking through as a prime drops to practically zero (less than the chance of a cosmic ray causing a computer hardware error).

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Saturday, October 3, 2026

Identifying Acute vs Obtuse triangles

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Evolution of Shapes

Tuesday, September 29, 2026

Proof by Completing the Square — the formula for for solving quadratic equationss

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Monday, September 28, 2026

Sec2 Finding roots of quadratic equation by completing the square

____ Basically, for this method of “completing the square” of a quad equation “x^2 + bx + c = 0”, purposely choose to square (x + b/2) then, minus away the (b/2)^2