Q.If $a$ and $b$ are positive integers and $\dfrac{a-b}{6.25} = \dfrac{8}{2.5}$

Integer Comparison: What It Means and Why It Matters

Imagine you have two piles of stones. One pile has 5 stones, the other has 3. Without counting, you can see the first pile is bigger. That's comparison in the real world — you're asking "which is larger?" or "are they equal?"

Integer comparison is just that same question, but asked about whole numbers (…, -3, -2, -1, 0, 1, 2, 3, …). The only twist is that integers include negative numbers, so the "bigger" pile might actually be a debt or a temperature below zero.

The Intuition: The Number Line

Picture a horizontal line. In the middle is 0. To the right are positive numbers (1, 2, 3, …), getting larger as you go right. To the left are negative numbers (-1, -2, -3, …), getting more negative (smaller) as you go left.

So:

• 5 is to the right of 3 → 5 > 3
• -2 is to the right of -5 → -2 > -5
• 0 is to the right of -1 → 0 > -1

The Precise Statement

For any two integers $a$ and $b$:

• $a > b$ (a is greater than b) if $a$ lies to the right of $b$ on the number line.
• $a < b$ (a is less than b) if $a$ lies to the left of $b$ on the number line.
• $a = b$ if $a$ and $b$ are the same point on the number line.

Equivalently, in terms of subtraction:

• $a > b$ means $a - b$ is positive.
• $a < b$ means $a - b$ is negative.
• $a = b$ means $a - b = 0$.

Common Pitfalls