The number can be written down in any of the existing position numeral systems where the value of each numerical sign (figures or letters) depends on its position (category). Besides decimal, binary, hexadecimal and octal systems are most known. In a position numeral system it is possible to make arithmetic operations over numbers. Subtraction and addition are defined by rules of addition of digits and order of the basis. It is enough to apply the multiplication table in the corresponding numeral system to multiplication and division.
Instruction
1. All arithmetic operations with numbers in numeral systems are performed, starting with the younger category (from right to left). At any operation of number register so that extreme signs were precisely the friend under the friend on the right. Actions with one-digit numbers, that is, consisting of one sign, are made taking into account the numeral system basis. At word length of system N, its numbers have values from 0 to N-1. If the received values over N-1, then N-1 is subtracted from result, the rest registers in the current units and the following category is added to number.
2. At addition of the multidigit numbers (containing several numerical or letter marks in record) it is necessary to make in addition transfer at overflow of the category and to consider it at addition of the subsequent figures or signs of number. In a binary system with the basis 2 there are only two figures: 0 and 1. Overflow happens here at addition of units, at the same time in the younger category registers 0 and is added 1 to senior. Similarly in any other position numeral system, the corresponding basis is only considered.
3. Subtraction is made by already known rules of a loan of unit from the senior category. Subtracting two numbers in an octal system, for example, numbers 2743 and 1371, write down them the friend under the friend – from above reduced, from below subtracted, draw a horizontal line even below. From right to left take away units at first of the younger category, then the subsequent, etc. At subtraction from the 3rd 1 – as a result will be 2, further from 4 it is subtracted 7 and here it will be required to carry out a loan from the senior category. For this purpose to 4 add the basis of this numeral system – number 8, subtract number 7 from the turned-out value (8+4=12) – will remain 5, write down this result below the line.
4. In the subsequent, senior category from 7 subtract busy unit, there will be number 6. From it already take away the number standing below – 3. As a result will remain 3, write down result below the line. Carry out subtraction over the last numbers – 2-1=1 – the final result of operation in an octal system looks so: 1352.
5. Multiplication of multidigit binary numbers is made according to the special table according to the usual scheme applied and in a decimal system. The work of numbers is made by means of alternate multiplication of one-digit numbers, the corresponding record of results and their further addition by a column with shift.