User:Phlsph7/Arithmetic other types

Other types of arithmetic

Example of modular arithmetic using a clock: after adding 4 hours to 9 o'clock, the hand starts at the beginning again and points at 1 o'clock.

There are many other types of arithmetic. Modular arithmetic operates on a finite set of numbers. If an operation would result in a number outside this finite set then the number is adjusted back into the set, similar to how the hands of clocks start at the beginning again after having completed one cycle. The number at which this adjustment happens is called the modulus. For example, a regular clock has a modulus of 12. In the case of adding 4 to 9, this means that the result is not 13 but 1. The same principle applies also to other operations, such as subtraction, multiplication, and division.

Some forms of arithmetic deal with operations performed on mathematical objects other than numbers. Interval arithmetic describes operations on intervals. Intervals can be used to represent a range of values if one does not know the precise magnitude, for example, because of measurement errors. Interval arithmetic includes operations like addition and multiplication on intervals, as in $[1,2]+[3,4]=[4,6]$ and $[1,2]\times [3,4]=[3,8]$. It is closely related to affine arithmetic, which aims to give more precise results by performing calculations on affine forms rather than intervals. An affine form is a number together with error terms that describe how the actual magnitude may deviate from the number. Vector arithmetic and matrix arithmetic describe arithmetic operations on vectors and matrices, like vector addition and matrix multiplication.

For mental arithmetic, the calculations are performed exclusively in the mind without the use of external tools like pan and paper or electronic calculators. Instead, mental arithmetic makes use of visualization, memorization, and certain calculation techniques to solve arithmetic problems. One such technique is the compensation method. It consists in altering the numbers to make the calculation easier and then adjusting the result afterward. For example, instead of calculating $85-47$, one calculates $85-50$ which is easier because it uses a round number. In the next step, one adds $3$ to the result to compensate for the earlier adjustment. Mental arithmetic is often taught in primary education to train the numerical abilities of the students.

Arithmetic systems can be classified based on the numeral system they rely on. For example, decimal arithmetic describes arithmetic operations in the decimal system. Other examples are binary arithmetic, octal arithmetic, and hexadecimal arithmetic.

Compound unit arithmetic describes arithmetic operations performed on magnitudes with compound units. It involves additional operations to govern the transformation between single unit and compound unit quantities. For example, the operation of reduction is used to transform the compound quantity 1 h 90 min into the single unit quantity 150 min.

Non-Diophantine arithmetics are arithmetic systems that violate traditional arithmetic intuitions and include equations like $1+1=1$ and $2+2=5$. They can be employed to represent some real-world situations in modern physics and everyday life. For instance, the equation $1+1=1$ can be used to describe the observation that if one raindrop is added to another raindrop then they do not remain two separate entities but become one.

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