Magnitude Comparators#

A magnitude comparator decides the relationship between two binary numbers — whether A equals B, A is greater, or A is less. It answers in logic what a subtraction answers in arithmetic, and it is a distinct, common building block wherever hardware has to make a decision about a value.

Note: this is the digital comparator, comparing multi-bit numbers. It shares its name with the analog voltage comparator in Analog Helpers, which compares two continuous voltages and outputs a single bit — a different device for a different job.

Equality Is a Row of XNORs#

The simplest comparison is equality. Two numbers are equal exactly when they match in every bit, and “these two bits are the same” is precisely what an XNOR reports. So an equality comparator is one XNOR per bit position, ANDed together: A = B is true only when all the per-bit XNORs are 1. This is the direct use the XNOR page points to.

Greater and Less#

Deciding which is larger takes an ordered look. Working from the most-significant bit down, the numbers are equal until the first bit position where they differ; at that position, whichever number has the 1 is the larger, and lower bits no longer matter. That priority-from-the-top structure is what the greater-than and less-than logic implements. Equivalently, subtracting one from the other and inspecting the sign and borrow gives the same answer — comparison and subtraction are two views of the same operation.

Cascading#

The classic part is the 7485, a 4-bit magnitude comparator with A>B, A=B, and A<B outputs and matching cascade inputs. The cascade inputs let comparators be chained: a higher stage’s result dominates, and its A=B output hands the decision down to the next stage only when its own bits tie. Stringing several together compares words of any width — the same “resolve from the most-significant end” rule, now spanning chips. Magnitude comparison shows up in threshold and limit checks, address-range decoding, sorting, and any control decision that hinges on one value being larger than another.