There are also situations where addition is "understood", even though no symbol appears:
Addition is used to model many physical processes. Even for the simple case of adding natural numbers, there are many possible interpretations and even more visual representations.
Possibly the most fundamental interpretation of addition lies in combining sets:
A second interpretation of addition comes from extending an initial length by a given length:
The fact that addition is commutative is known as the "commutative law of addition" or "commutative property of addition". Some other binary operations are commutative, such as multiplication, but many others are not, such as subtraction and division.
Children are often presented with the addition table of pairs of numbers from 0 to 9 to memorize. Knowing this, children can perform any addition.
Adding two "1" digits produces a digit "0", while 1 must be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:
1 1 1 1 1 (carried digits) 0 1 1 0 1 + 1 0 1 1 1 ————————————— 1 0 0 1 0 0 = 36
Addition, along with subtraction, multiplication and division, is considered one of the basic operations and is used in elementary arithmetic.
Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition:
Tying these observations together, tropical addition is approximately related to regular addition through the logarithm:
Addition is also used in the musical set theory. George Perle gives the following example: “do-mi, re-fa♯, mi♭-sol are different requirements of one interval... or other type of equality... and are connected with the axis of symmetry. Do-mi belongs to the family of symmetrically connected dyads as it is shown further:”
Axis of the pitches are italicized, the axis is defined with the pitch category.
Thus, do-mi is a part of an interval family-4 and a part of sum family -2 (at G♯ = 0).
The total number of successive dyads of a tonal range in Lyric Suite is 11
Axis of the pitches are italicized, the axis is defined by the dyads (interval 1).