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Example of addition with carry. The black numbers are the addends, the green number is the carry, and the blue number is the sum. In the rightmost digit, the addition of 9 and 7 is 16, carrying 1 into the next pair of the digit to the left, making its addition 1 + 5 + 2 = 8. Therefore, 59 + 27 = 86.
For example, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to musical set theory. The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as the " exclusive or " function.
Many of these problems are studied using the tools from the Hardy-Littlewood circle method and from sieve methods. For example, Vinogradov proved that every sufficiently large odd number is the sum of three primes, and so every sufficiently large even integer is the sum of four primes.
The term "number bond" is also used to refer to a pictorial representation of part-part-whole relationships, often found in the Singapore mathematics curriculum. Number bonds consist of a minimum of 3 circles that are connected by lines. The “whole” is written in the first circle and its “parts” are written in the adjoining circles.
It is part of the standard algorithm to add numbers together by starting with the rightmost digits and working to the left. For example, when 6 and 7 are added to make 13, the "3" is written to the same column and the "1" is carried to the left. When used in subtraction the operation is called a borrow.
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Partition into cliques is the same problem as coloring the complement of the given graph. A related problem is to find a partition that is optimal terms of the number of edges between parts. [3]: GT11, GT12, GT13, GT14, GT15, GT16, ND14 Grundy number of a directed graph. [3]: GT56 Hamiltonian completion [3]: GT34
We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c.. For the base case c = 0, (a + b) + 0 = a + b = a + (b + 0)Each equation follows by definition [A1]; the first with a + b, the second with b.
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