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Colon Equals — Definition, Formula & Examples

Colon equals (:=) is a symbol meaning "is defined as" or "is assigned the value of." It signals that the left-hand side is being introduced as a new name or notation for the expression on the right.

The symbol :=:= denotes a definitional equality, where the quantity or expression on the left side of :=:= is defined to be equal to the expression on the right. Unlike the standard equality sign ==, which asserts that two independently meaningful expressions have the same value, :=:= establishes the meaning of one side in terms of the other.

Key Formula

x:=expressionx := \text{expression}
Where:
  • xx = The new symbol or variable being defined
  • expression\text{expression} = The known expression that gives x its meaning

How It Works

When you see A:=BA := B, read it as "AA is defined to be BB." The colon sits on the side of the symbol being defined. Some authors reverse the convention and write B=:AB =: A to define AA, placing the colon next to the new term. In programming languages like Pascal and Go, :=:= serves as the assignment operator, which inspired its mathematical usage. You will encounter :=:= most often in proofs, formal definitions, and algorithm descriptions where distinguishing a definition from a derived equation matters.

Worked Example

Problem: Define a function f(x) that gives the square of a number plus one, then evaluate it at x = 3.
Define: Use the colon-equals symbol to introduce the definition of f.
f(x):=x2+1f(x) := x^2 + 1
Evaluate: Substitute x = 3 into the defined expression.
f(3)=32+1=10f(3) = 3^2 + 1 = 10
Answer: After defining f(x):=x2+1f(x) := x^2 + 1, we find f(3)=10f(3) = 10. The :=:= told us this was a definition, not an equation to solve.

Why It Matters

In proof-based courses like real analysis and abstract algebra, distinguishing definitions from derived equalities prevents logical confusion. The :=:= symbol makes it immediately clear that a line is establishing new notation rather than claiming a result that needs justification.

Common Mistakes

Mistake: Treating := and = as interchangeable.
Correction: The symbol == asserts that two expressions are equal (which may need proof), while :=:= declares a definition. Writing a=b+1a = b + 1 in a proof context may be ambiguous — the reader cannot tell if this is given, derived, or a definition. Use a:=b+1a := b + 1 when you are defining aa.

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