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other answers are good, just throwing another one out there for further consideration/insight with a CS flavor/twist. one can construct an FSM, a finite state machine, that can compare two binary numbers of any length, starting pairwise from the most significant bits and working to least significant bit (LSB). it can also be used to conceptualize the digital comparator given in another answer, yet the FSM does not require finite length binary numbers. it can even work on integers with binary fractions after the LSB. it has an inductive and recursive flavor and can be proven correct by simple induction. it runs as follows:

• input the top two binary digits as a pair (a,b)
• if a=1 and b=0 the left number is larger.
• if a=0 and b=1, the right number is larger.
• otherwise the numbers are "equal up til now", advance to next pair.

in other words the largest number is the one with the first occurrence of a bit that is one and the other is zero, after an initial run of zero or more identical 1's. a finite-length digital comparator made of gates or 1-bit comparators can be seen as based on fixing the length of this FSM operation to some fixed number of bits. (yes there is a strong correspondence between all finite circuits and "fixing the length" of FSM computations.)

this may seem like a theoretical exercise but actually, the logic in software to represent arbitrary precision numbers operates something analogous to this FSM, except encoded in a computer loop that can be seen as looping over the steps of the FSM.

also, lets reasonably interpret/generalize this question as not limited to integers. the question refers to integers but the title only refers to numbers. surprisingly nobody else mentioned floating point arithmetic so far.

basically that works by comparing the exponent and the mantissa where a number is in the form $a \times 10^b$, $a$ the mantissa, b the exponent. the mantissa can be normalized to a number where the first digit is always nonzero. then to compare two numbers the logic first compares the exponents $b$, and if they are unequal, it can return a result without comparing the mantissas (using say the comparator circuit). if the exponents are equal, it compares the mantissas.

other answers are good, just throwing another one out there for further consideration/insight with a CS flavor/twist. one can construct an FSM, a finite state machine, that can compare two binary numbers of any length, starting pairwise from the most significant bits and working to least significant bit (LSB). it can also be used to conceptualize the digital comparator given in another answer, yet the FSM does not require finite length binary numbers. it can even work on integers with binary fractions after the LSB. it has an inductive and recursive flavor and can be proven correct by simple induction. it runs as follows:

• input the top two binary digits as a pair (a,b)
• if a=1 and b=0 the left number is larger.
• if a=0 and b=1, the right number is larger.
• otherwise the numbers are "equal up til now", advance to next pair.

in other words the largest number is the one with the first occurrence of a bit that is one and the other is zero, after an initial run of zero or more identical 1's. a finite-length digital comparator made of gates or 1-bit comparators can be seen as based on fixing the length of this FSM operation to some fixed number of bits. (yes there is a strong correspondence between all finite circuits and "fixing the length" of FSM computations.)

this may seem like a theoretical exercise but actually, the logic in software to represent arbitrary precision numbers operates something analogous to this FSM, except encoded in a computer loop that can be seen as looping over or simulating the steps of the FSM (an efficient implementation might track via an index the location of the MSB).

also, lets reasonably interpret/generalize this question as not limited to integers. the question refers to integers but the title only refers to numbers. surprisingly nobody else mentioned floating point arithmetic so far.

basically that works by comparing the exponent and the mantissa where a number is in the form $a \times 10^b$, $a$ the mantissa, b the exponent. the mantissa can be normalized to a number where the first digit is always nonzero. then to compare two numbers the logic first compares the exponents $b$, and if they are unequal, it can return a result without comparing the mantissas (using say the comparator circuit). if the exponents are equal, it compares the mantissas.

other answers are good, just throwing another one out there for further consideration/insight with a CS flavor/twist. one can construct an FSM, a finite state machine, that can compare two binary numbers of any length, starting pairwise from the most significant bits and working to least significant bit (LSB). it can also be used to conceptualize the digital comparator given in another answer, yet the FSM does not require finite length binary numbers. it can even work on integers with binary fractions after the LSB. it has an inductive and recursive flavor and can be proven correct by simple induction. it runs as follows:

• input the top two binary digits as a pair (a,b)
• if a=1 and b=0 the left number is larger.
• if a=0 and b=1, the right number is larger.
• otherwise the numbers are "equal up til now", advance to next pair.

in other words the largest number is the one with the first occurrence of a bit that is one and the other is zero, after an initial run of zero or more identical 1's. a finite-length digital comparator made of gates or 1-bit comparators can be seen as based on fixing the length of this FSM operation to some fixed number of bits. (yes there is a strong correspondence between all finite circuits and "fixing the length" of FSM computations.)

this may seem like a theoretical exercise but actually, the logic in software to represent arbitrary precision numbers operates something analogous to this FSM, except encoded in a computer loop that can be seen as looping over the steps of the FSM.

also, lets reasonably interpret/generalize this question as not limited to integers. the question refers to integers but the title only refers to numbers. surprisingly nobody else mentioned floating point arithmetic so far.

basically that works by comparing the exponent and the mantissa where a number is in the for $a \times 10^b$, $a$ the mantissa, b the exponent. the mantissa can be normalized to a number where the first digit is always nonzero. then to compare two numbers the logic first compares the exponents $b$, and if they are unequal, it can return a result without comparing the mantissas (using say the comparator circuit). if the exponents are equal, it compares the mantissas.

other answers are good, just throwing another one out there for further consideration/insight with a CS flavor/twist. one can construct an FSM, a finite state machine, that can compare two binary numbers of any length, starting pairwise from the most significant bits and working to least significant bit (LSB). it can also be used to conceptualize the digital comparator given in another answer, yet the FSM does not require finite length binary numbers. it can even work on integers with binary fractions after the LSB. it has an inductive and recursive flavor and can be proven correct by simple induction. it runs as follows:

• input the top two binary digits as a pair (a,b)
• if a=1 and b=0 the left number is larger.
• if a=0 and b=1, the right number is larger.
• otherwise the numbers are "equal up til now", advance to next pair.

in other words the largest number is the one with the first occurrence of a bit that is one and the other is zero, after an initial run of zero or more identical 1's. a finite-length digital comparator made of gates or 1-bit comparators can be seen as based on fixing the length of this FSM operation to some fixed number of bits. (yes there is a strong correspondence between all finite circuits and "fixing the length" of FSM computations.)

this may seem like a theoretical exercise but actually, the logic in software to represent arbitrary precision numbers operates something analogous to this FSM, except encoded in a computer loop that can be seen as looping over the steps of the FSM.

also, lets reasonably interpret/generalize this question as not limited to integers. the question refers to integers but the title only refers to numbers. surprisingly nobody else mentioned floating point arithmetic so far.

basically that works by comparing the exponent and the mantissa where a number is in the form $a \times 10^b$, $a$ the mantissa, b the exponent. the mantissa can be normalized to a number where the first digit is always nonzero. then to compare two numbers the logic first compares the exponents $b$, and if they are unequal, it can return a result without comparing the mantissas (using say the comparator circuit). if the exponents are equal, it compares the mantissas.

other answers are good, just throwing another one out there for further consideration/insight with a CS flavor/twist. one can construct an FSM, a finite state machine, that can compare two binary numbers of any length, starting pairwise from the most significant bits and working to least significant bit (LSB). it can also be used to conceptualize the digital comparator given in another answer, yet the FSM does not require finite length binary numbers. it can even work on integers with binary fractions after the LSB. it has an inductive and recursive flavor and can be proven correct by simple induction. it runs as follows:

• input the top two binary digits as a pair (a,b)
• if a=1 and b=0 the left number is larger.
• if b=1 and a=0, the right number is larger.
• otherwise the numbers are "equal up til now", advance to next pair.

in other words the largest number is the one with the first occurrence of a bit that is one and the other is zero, after an initial run of zero or more identical 1's. a finite-length digital comparator made of gates or 1-bit comparators can be seen as based on fixing the length of this FSM operation to some fixed number of bits. (yes there is a strong correspondence between all finite circuits and "fixing the length" of FSM computations.)

other answers are good, just throwing another one out there for further consideration/insight with a CS flavor/twist. one can construct an FSM, a finite state machine, that can compare two binary numbers of any length, starting pairwise from the most significant bits and working to least significant bit (LSB). it can also be used to conceptualize the digital comparator given in another answer, yet the FSM does not require finite length binary numbers. it can even work on integers with binary fractions after the LSB. it has an inductive and recursive flavor and can be proven correct by simple induction. it runs as follows:

• input the top two binary digits as a pair (a,b)
• if a=1 and b=0 the left number is larger.
• if a=0 and b=1, the right number is larger.
• otherwise the numbers are "equal up til now", advance to next pair.

in other words the largest number is the one with the first occurrence of a bit that is one and the other is zero, after an initial run of zero or more identical 1's. a finite-length digital comparator made of gates or 1-bit comparators can be seen as based on fixing the length of this FSM operation to some fixed number of bits. (yes there is a strong correspondence between all finite circuits and "fixing the length" of FSM computations.)

this may seem like a theoretical exercise but actually, the logic in software to represent arbitrary precision numbers operates something analogous to this FSM, except encoded in a computer loop that can be seen as looping over the steps of the FSM.

also, lets reasonably interpret/generalize this question as not limited to integers. the question refers to integers but the title only refers to numbers. surprisingly nobody else mentioned floating point arithmetic so far.

basically that works by comparing the exponent and the mantissa where a number is in the for $a \times 10^b$, $a$ the mantissa, b the exponent. the mantissa can be normalized to a number where the first digit is always nonzero. then to compare two numbers the logic first compares the exponents $b$, and if they are unequal, it can return a result without comparing the mantissas (using say the comparator circuit). if the exponents are equal, it compares the mantissas.