Numbering System Problem Solved

 

Numbering System Problem Solved

Problem Carry out the following conversions:a) 1238 to the binary system.B) 10101012 to the octal system.C) BACA16 to the binary system.D) 10101012 to the hexadecimal system.E) 1316310 to the binary system.Sug. Use successive divisions and polynomial decomposition. T. Solution We present 2 ways to solve this problem. A) First Way: Let's say that this is the trivial way to carry out this process. We convert the number 1238 to a number in base 10 (the one we use), through the polynomial decomposition of the following form:\( 123_8 = 1 \times 8^2 + 2\times 8^1 + 3 \times 8^0 \)\( \rightarrow 1 \times 64 + 2\times 8 + 3 \times 1 \ \rightarrow 83\)Then we convert the number 83 to a number in base 2 (binary) through successiv divisions:

We proceed to write the numbers that are enclosed in a square, starting with the number that is on the left side and ending with the one on the right side, thus obtaining the number in base 2, that is:\( 83 = 83_{10} = 1010011_2 \)Then we will say that the number 123 in base 8 is equivalent to the number 83 in base 10 and these in turn are equivalent to the number 1010011 in base 2.Second FormLet's examine the following table, it shows us numbers in the decimal system (base 10), in the octal system (base 8) and in the binary system (base 2).

We see the equivalences between the values ​​of the different numbering systems, for example the number 5 in base 10 is equivalent to the number 5 in base 8 (octal) and in turn these are equivalent to the number 101 in base 2 (binary). Whenever we want converting a number from base 2 to base 8 or vice versa, it is convenient to express base 2 numbers (binary system) as 3-digit numbers as in the tabl