XXIIVV

Arithmetics is study of numbers, especially the properties of the traditional operations on them.

Peasant Multiplication

In the first column, divide the first number by 2, dropping the remainder if any, until 1 is reached. In the second column, write the numbers obtained by successive multiplication by 2. The answer is found by adding the numbers in the doubling column with odd numbers in their first column.

64 x 61
6461
32122
16244
8488
4976
21952
13904+3904
Result: 3904
61 x 64
6164+64
30128
15256+256
7512+512
31024+1024
12048+2048
Result: 3904

Addition of 5

When adding 5 to a digit greater than 5, it is easier to first subtract 5 and then add 10.

7 + 5 = 12.
Also 7 - 5 = 2; 2 + 10 = 12.

Subtraction of 5

When subtracting 5 from a number ending with a a digit smaller than 5, it is easier to first add 5 and then subtract 10.

23 - 5 = 18.
Also 23 + 5 = 28; 28 - 10 = 18.

Division by 5

Similarly, it's often more convenient instead to multiply first by 2 and then divide by 10.

1375/5 = 2750/10 = 275.

Multiplication by 5

It's often more convenient instead of multiplying by 5 to multiply first by 10 and then divide by 2.

137×5 = 1370/2 = 685.

Division by 5

Similarly, it's often more convenient instead to multiply first by 2 and then divide by 10.

1375/5 = 2750/10 = 275.

Division/multiplication by 4

Replace either with a repeated operation by 2.

124/4 = 62/2 = 31. Also,
124×4 = 248×2 = 496.

Division/multiplication by 25

Use operations with 4 instead.

37×25 = 3700/4 = 1850/2 = 925.

Division/multiplication by 8

Replace either with a repeated operation by 2.

124×8 = 248×4 = 496×2 = 992.

Division/multiplication by 125

Use operations with 8 instead.

37×125 = 37000/8 = 18500/4 = 9250/2 = 4625.
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binary

Binary numbers are a base 2 numeral system.

A binary number is a number expressed in the base-2 numeral system, which uses only two symbols: 0 and 1. Each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices.

 
—Two of Leibniz's binary calculation examples

To explore binary logic, see Noton or Der Papiercomputer.

reverse polish

In Reverse Polish Notation, the operators follow their operands.

In RPN calculators, no equals key is required to force computation to occur. To learn more about a programming language using RPN at its core, see Forth. To find a simple RPN implementation and playground, see Firth.