How many integers are there between two successive integers?
Sounds like a trick question, right? You picture the number line, pick 4 and 5, and then stare at the empty gap between them. The answer is “zero,” but getting there involves a few mental detours that most people never think about.
If you’ve ever argued with a kid about “what’s in between 7 and 8?” or tried to explain the idea in a math class, you know the conversation can spin into a surprisingly deep rabbit hole. Let’s untangle it together.
What Is “Between Two Successive Integers”
When we talk about successive integers we mean two whole numbers that follow each other directly on the number line—no other whole numbers in between. In everyday language you might hear “consecutive numbers” or “adjacent integers.”
Think of the integers as the stepping stones you’d use to cross a river. So naturally, the question “how many integers are there between 4 and 5? Each stone is a whole number, and the water between them is the “non‑integer” space. ” is asking how many stepping stones you can place inside that water. The short answer: none Which is the point..
Formal definition (without the textbook feel)
An integer n is a whole number that can be written without fractions or decimals, positive, negative, or zero. Worth adding: two integers a and b are successive if b = a + 1. The “between” part means any integer k that satisfies a < k < b. Because b = a + 1, there is no integer that can fit strictly between them Which is the point..
Why It Matters / Why People Care
You might wonder why anyone would waste time on a question that seems obvious. The truth is, the idea crops up in more places than you think.
- Programming bugs – When you write a loop that iterates from i = 1 to i < n, you’re implicitly counting the integers between two bounds. Misunderstanding “between” can cause off‑by‑one errors, the bane of every coder.
- Number theory – The concept of “no integer between successive integers” underpins the well‑ordering principle, which says every non‑empty set of positive integers has a least element. That principle is a building block for proofs about divisibility, primes, and more.
- Real‑world measurements – If you’re measuring in whole centimeters, you can’t have a “half‑centimeter” integer between 10 cm and 11 cm. Knowing the gap is empty helps you set tolerances correctly.
In practice, the fact that there’s nothing in that gap gives us a clean, predictable structure for the whole number system. It’s the reason you can count “one, two, three…” without worrying about hidden steps.
How It Works
Let’s break the logic down step by step, so you can see why the answer is zero and not “maybe one” or “infinitely many.”
1. Identify the two integers
Pick any integer a. Think about it: its successor is a + 1. As an example, let a = 12. Then the successive integer is 13 No workaround needed..
2. Write the inequality that defines “between”
We’re looking for integers k such that:
a < k < a + 1
Plugging in the numbers:
12 < k < 13
3. Consider the properties of integers
An integer can only be a whole number: …, 10, 11, 12, 13, 14, … There is no whole number that is greater than 12 and less than 13. The next whole number after 12 is already 13, which violates the strict “less than” part.
4. Generalize the proof
Assume, for contradiction, that there exists an integer k with a < k < a + 1. Because k is an integer, the difference k − a must be a positive integer (at least 1). But k − a is also less than 1 (since k < a + 1). The only integer that is both ≥ 1 and < 1 is impossible. Hence no such k exists.
That’s the core reasoning. It works for any pair of successive integers, no matter how large or how negative.
Common Mistakes / What Most People Get Wrong
Even though the answer is “zero,” people stumble for a few predictable reasons It's one of those things that adds up..
Mistake #1: Including the endpoints
Some folks count the two given integers as “between.” Remember, “between” in mathematics is strict: it excludes the endpoints. If you want to include them, you’d say “from a to b inclusive.
Mistake #2: Confusing with real numbers
If you switch the domain from integers to real numbers, the answer changes dramatically. Day to day, between 4 and 5 there are infinitely many real numbers (4. 1, 4.001, π, etc.Think about it: ). The key is the word integer—the set is discrete, not continuous Turns out it matters..
Mistake #3: Assuming fractions count as integers
A common slip is to think “½” or “3.But 7” are “in between” because they look like numbers. They’re not integers, so they don’t count for this particular question.
Mistake #4: Over‑thinking the “successive” part
Sometimes people pick numbers that aren’t actually successive, like 4 and 6, and then wonder why there’s a missing integer. The phrase “successive” is the guardrail; if the gap is larger than one, there will be integers in between Which is the point..
Practical Tips / What Actually Works
If you’re teaching this concept, writing code, or just need a quick mental check, keep these pointers handy.
- Use the “+1” test – Whenever you have two integers, subtract the smaller from the larger. If the difference is exactly 1, you have successive integers and the answer is zero.
- Write it out – Jot the numbers on paper: 7 | 8. The vertical bar helps you see the empty space.
- Remember the strict inequality – Say the phrase “strictly between” out loud. It reinforces that the endpoints are off‑limits.
- Check with a simple loop (programming tip):
a = 9
b = a + 1
count = 0
for k in range(a+1, b):
count += 1
print(count) # prints 0
If the loop never runs, you’ve confirmed there’s nothing in the gap Which is the point..
- Teach with a number line visual – Draw a line, mark the two integers, and shade the region between them. The empty space is a powerful visual cue for learners.
FAQ
Q: Are there ever any integers between two successive integers if we allow negative numbers?
A: No. The definition works the same way for negative numbers. Between –3 and –2 there’s still no integer.
Q: What about zero? Is there an integer between –1 and 0?
A: No. –1 and 0 are successive; the gap contains no whole numbers Not complicated — just consistent..
Q: If I’m dealing with modular arithmetic, does the “no integer between” rule still hold?
A: Absolutely. Modulo just wraps the number line around a circle, but the distance between a number and its successor is still one unit, leaving no integer in between.
Q: Can I use this fact to prove that the set of integers is infinite?
A: Indirectly, yes. Because you can always add 1 to any integer and get a new one, there’s no “last” integer. The emptiness of the gap guarantees you can keep stepping forward forever.
Q: How does this relate to prime numbers?
A: Prime gaps are measured in terms of successive prime numbers, not successive integers. Between 7 and 11 there are two integers (8, 9, 10) that are not prime. The “no integer between successive integers” idea tells us the smallest possible prime gap is 2 (the twin primes 11 and 13, for example) That's the part that actually makes a difference..
Wrapping It Up
So, how many integers are there between two successive integers? Zero. It’s a tiny fact, but it’s a cornerstone of how we think about whole numbers, loops in code, and even the proof that there’s no biggest integer The details matter here..
Next time you hear someone ask, “What’s between 4 and 5?” you can smile, point to the empty space on the number line, and say, “Nothing, literally nothing.” It’s a simple answer that carries a lot of mathematical weight.
