Invariance Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Invariance.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A property of a mathematical object that remains unchanged when the object undergoes a particular transformation or operation.

What stays the same when things change? That's often the key.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Finding what stays fixed under a transformation reveals the deepest structure — invariants are the "bones" of the mathematical object.

Common stuck point: Invariance is always relative to a specific transformation — area is invariant under rotation but not under scaling.

Sense of Study hint: Apply the transformation to a specific example, then compare before and after. List what changed and what stayed the same.

Worked Examples

Example 1

easy
Show that the sum of the digits of a multiple of 9 is always a multiple of 9. Verify with n = 198 and n = 729.

Solution

  1. 1
    Any integer n can be written as n = \sum_i a_i \cdot 10^i where a_i are digits. Since 10 \equiv 1 \pmod{9}, we get n \equiv \sum_i a_i \pmod{9}.
  2. 2
    So 9 \mid n \Leftrightarrow 9 \mid (\text{sum of digits}) — the divisibility by 9 is an invariant property shared by n and its digit sum.
  3. 3
    Check n=198: digit sum = 1+9+8 = 18, which is a multiple of 9. And 198 = 9 \times 22. Confirmed.
  4. 4
    Check n=729: digit sum = 7+2+9 = 18, multiple of 9. And 729 = 9 \times 81. Confirmed.

Answer

9 \mid n \Leftrightarrow 9 \mid (\text{digit sum of }n)
An invariant is a property preserved across transformations. Here, taking digit sums preserves divisibility by 9 because of how 10 behaves modulo 9.

Example 2

medium
A sequence starts at 1 and each step either doubles the value or adds 3. Show that the parity (odd/even) of the value changes predictably and identify an invariant.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Show that the expression x^2 + y^2 is invariant under the transformation (x, y) \mapsto (-x, -y).

Example 2

medium
In a game, you start with the number 6. Each move you may subtract 1 or divide by 2 (if even). Show that the quantity n \pmod{3} is not always preserved and find an invariant that is preserved.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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