The variety of numeral systems in mathematics is explained by various origin of theories of numbers, both territorial, and applied. For example, with development of computers and other technical means the big distribution was received by a relatively young binary system. Quinary also is position, it was an account basis in the ancient tribe of the Maya.
Instruction
1. A numeral system – the integral part of the mathematical theory which is responsible for symbolical record of numbers. Each system has own arithmetics, set of actions: addition, multiplication, division and multiplication.
2. The basis of a quinary system is figure 5. Respectively, this number represents one category, for example, 132 in a quinary system represents 2•5^0 + 3•5 ¹ + 1•5² = 2 + 15 + 25 = 42 in denary.
3. To transfer number to a quinary system from any other position numeral system, use method of consecutive division. You divide required number into 5, writing down the intermediate remains upside-down, i.e. from right to left.
4. Begin with a decimal system. Translate number 69:69/5 = 13 → 4 in the rest; 13/5 = 2 → 3;2/5 = 0 → 2.
5. So, number 234 turned out. Check result: 234 = 4•1 + 3•5 + 2•25 = 69.
6. It is possible to make the translation of number of any other system in two ways: or the same consecutive division, or using an intermediate system most of which convenient option will be decimal. Despite existence of an additional stage, the second method faster and exact as does not assume actions of unusual arithmetics. For example, lead octal number 354 to a quinary look.
7. Use the first way: 354/5 = 57 → 1 in the rest; 57/5 = 11 → 2;11/5 = 1 → 4;1/5 = 0 → 1.
8. Inconveniently, isn't that so? All the time it is necessary to remember that the divisible number has the word length equal 8, but not 10 though the eye which is swept together on decimal operations it is deceptive perceives it quite so. Now apply the second way: Pass to a decimal look: 354 = 4•1 + 5•8 +3·64 = 236.
9. Make the habitual translation: 236/5 = 47 → 1;47/5 = 9 → 2;9/5 = 1 → 4;1/5 = 0 → 1.
10. Write down result: 354_8 = 1421_5. Check: 1421=1·1+2*5+4 · 25+1 · 125=236.