A kosarad üres
Number systems
Back in the Archean Eon signs were used to write down numbers. If large numbers were needed, new signs were introduced. In the developed cultures of the Antiquity besides putting down large numbers also operations with large numbers became necessary. The numbers were grouped and new signs were introduced for each group. Depending on the number of numbers we create a new group out of, we talk about different number systems.
The base-five number system is still used by some Indian tribes living in South America. They count as follows: one, two, three, four, hand, hand and one, hand and two, etc.
The base-six number system is used by some tribes living in Northwest Africa; it is used mixed with base-twelve number system. Signs indicating the latter one can also be realised in the European cultures. It is enough to think of the months of the year or the scale of the clock.
The base-twenty number system was used in the Mayan and Celtic cultures. In Mexico and Middle America it is still used in astronomy.
The Babylonians counted in base-sixty number system, this is where the \latex{60} minutes of an hour, the \latex{60} seconds of a minute and the angle measuring system stem from.
The Roman numerals show the mixture of the base-five and the base-ten number systems. These signs have been used in Europe for centuries, although doing the operations with them is quite complicated.
The easiest way for calculation is using our hand with the fingers; it explains why the decimal number system has spread generally.
The first signs of the numerals with place values can be found in the Hindu culture of the Antiquity. There the written digits represent the multiples of the powers of the base of the number system.
For example \latex{312} in the decimal number system contains the following sum: \latex{3\times 10^{2}+1\times10+2}.
The numerals with place values in the decimal number system spread in Europe through Arabic scientists, and it could squeeze out the usage of Roman numerals only in several centuries. It was a significant innovation to introduce \latex{0} as a placeholder number. In other cultures of the Antiquity \latex{0} was not considered as a number, it did not even have a corresponding sign.
The base of the number system determines the number of digits which can be used. For example in the base-five number system only five digits \latex{\left(0;1;2;3;4\right)} can appear.
Example 1
Let us convert number \latex{2,103_{5}} to the decimal number system. (The index in the subscript means the base of the number system.)
Solution
\latex{2,103_{5}=2\times5^{3}+1\times5^{2}+0\times 5+3=278}.
In the base-ten (decimal) number system the base is not denoted.
The table of place values in base-\latex{5} number system
\latex{5^{4}}
\latex{5^{3}}
\latex{5^{2}}
\latex{5^{1}}
\latex{5^{0}=1}
\latex{2}
\latex{1}
\latex{0}
\latex{3}
Example 2
How can \latex{821} be written in the base-six number system?
Solution
Let us make groups of \latex{6}, i.e. let us divide the number by \latex{6}.
\latex{821\div6=136}, and the remainder is \latex{5} ones.
\latex{821\div6=136}, and the remainder is \latex{5} ones.
Let us make new groups of \latex{6} from the \latex{136} pieces of groups of \latex{6}.
\latex{136\div 6 = 22}, and the remainder is \latex{4} groups of \latex{6}.
\latex{136\div 6 = 22}, and the remainder is \latex{4} groups of \latex{6}.
Let us make groups of \latex{6} from the \latex{22} pieces of groups of \latex{36}.
\latex{22\div 6 = 3}, and the remainder is \latex{4} groups of \latex{36}.
\latex{22\div 6 = 3}, and the remainder is \latex{4} groups of \latex{36}.
It is not possible to make groups of \latex{6} from the \latex{3} pieces of groups of \latex{216\div 3\div 6 = 0}, and the remainder is \latex{3} groups of \latex{216}.
dividing the number
by \latex{ 6 }
by \latex{ 6 }
Is the quotient
greater than \latex{ 0 }?
greater than \latex{ 0 }?
dividing
the quotient
by \latex{ 6 }
the quotient
by \latex{ 6 }
the remainders read
backwards give the number
in base-\latex{ 6 } number system
backwards give the number
in base-\latex{ 6 } number system
t
f
Thus \latex{816} ordered in the table of place values in the base-\latex{6} number system:
groups
number of groups
\latex{36=6^{2}}
\latex{216=6^{3}}
\latex{6=6^{1}}
\latex{6^{0}=1}
\latex{3}
\latex{4}
\latex{4}
\latex{5}
i.e. \latex{821 = 3,445_6}.
The sum form of the number with place values in the base-\latex{6} number system:
\latex{3,445_6=\underbrace{3\times 216+4\times 36+4\times6}_{\text{is divisible by 6}}+}\latex{\underbrace{5}_{\text{last digit}}} \latex{\times 1}.
It can be seen that the sum without the last digit is divisible by \latex{6}, thus it is divisible by \latex{2} and \latex{3} too. So the divisibility by \latex{6, 2} and \latex{3} in the base-\latex{6} number system can be decided based on the last digit. E.g. \latex{3,440_{6}} and \latex{3,443_{6}} are divisible by \latex{3}, but \latex{3,441_{6}} is not divisible by \latex{3}.
Example 3
Which numbers of the base-\latex{6} number system are divisible by \latex{5}?
Solution
Let us consider the following example:
\latex{3142_6=3\times 216+1\times 36+4\times 6+2\times 1=} \latex{\textcolor{#FFFFFF}{31426}=3\times \left(215+1\right)+1\times \left(35+1\right)+4\times \left(5+1\right)+2\times 1=} \latex{\textcolor{#FFFFFF}{31426}=\underbrace{3\times 215+1\times 35+4\times 5}_{\text{is divisible by 5} \atop }+\underbrace{3+1+4+2}_{\text{the sum of the } \atop \text{digits is divisible by 5}}}.
By generalising the example we get that a number written in the base-\latex{6} number system is divisible by \latex{5} if and only if the sum of its digits is divisible by \latex{5}.
Example 4
Let us create the table of addition and multiplication in base-\latex{5} number system, then with the help of the tables let us do the following operations:
- \latex{32,102_{5}+{14,213_5}};
- \latex{3,201_{5}\times {32_5}}.
Solution
\latex{\begin{array}{r} 32,102_5 \\+14,213_5\\\hline101,320_5\end{array}}
a) Let us write the numbers
to add below each other.
When adding pay attention
to carry the remainder.
to add below each other.
When adding pay attention
to carry the remainder.
b) Let us act similarly when
doing the multiplication.
doing the multiplication.
\latex{\begin{array}{l} \underline{3,201_5} \times32_5\\ 20,103\end{array}}
\latex{\begin{array}{l}11,402\\\overline{212,432_5}\end{array}}
table of addition and multiplication in base-\latex{5} number system
\latex{0}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{0}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{0}
\latex{1}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{2}
\latex{3}
\latex{4}
\latex{10}
\latex{2}
\latex{3}
\latex{4}
\latex{10}
\latex{11}
\latex{3}
\latex{4}
\latex{10}
\latex{11}
\latex{12}
\latex{4}
\latex{10}
\latex{11}
\latex{12}
\latex{13}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{1}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
\latex{2}
\latex{3}
\latex{4}
\latex{2}
\latex{4}
\latex{11}
\latex{13}
\latex{3}
\latex{11}
\latex{14}
\latex{22}
\latex{4}
\latex{13}
\latex{22}
\latex{31}
Example 5
One piece of a \latex{1} \latex{ kg }, a \latex{2} \latex{ kg }, a \latex{4} \latex{ kg }, an \latex{8} \latex{ kg } and a \latex{16} \latex{ kg } weight and a balancing scale are available. Objects with what mass can be measured if we place the known weights into one pan?
Solution
Obviously we can only measure integer kilograms, but up to \latex{31} \latex{ kg } we can measure anything since:
\latex{\bold{31}=16+8+4+2+1}; \latex{\bold{30}=16+8+4+2}; \latex{\bold{29}=16+8+4+1};
\latex{\bold{28}=16+8+4}; \latex{\bold{27}=16+8+2+1}; \latex{\bold{26}=16+8+2}; \latex{\bold{25}=16+8+1};
\latex{\bold{24}=16+8}; \latex{\bold{23}=16+4+2+1}; \latex{\bold{22}=16+4+2}; \latex{\bold{21}=16+4+1};
\latex{\bold{20}=16+4}; \latex{\bold{19}=16+2+1}; \latex{\bold {18}=16+2}; \latex{\bold{17}=16+1}; …; \latex{\bold{3}=2+1};
\latex{\bold{2}=2}; \latex{\bold{1}=1}.
\latex{\bold{28}=16+8+4}; \latex{\bold{27}=16+8+2+1}; \latex{\bold{26}=16+8+2}; \latex{\bold{25}=16+8+1};
\latex{\bold{24}=16+8}; \latex{\bold{23}=16+4+2+1}; \latex{\bold{22}=16+4+2}; \latex{\bold{21}=16+4+1};
\latex{\bold{20}=16+4}; \latex{\bold{19}=16+2+1}; \latex{\bold {18}=16+2}; \latex{\bold{17}=16+1}; …; \latex{\bold{3}=2+1};
\latex{\bold{2}=2}; \latex{\bold{1}=1}.
It corresponds to expressing the numbers in the base-\latex{2} number system. If a digit in the base-\latex{2} number system is \latex{1}, then the weight corresponding to it should be used, if it is \latex{0}, then it should not be used.
Exercises
{{exercise_number}}. Convert the following numbers to the decimal number system.
- \latex{34,056_8}
- \latex{10,111,101_2}
- \latex{22,302_5}
{{exercise_number}}. Which number is greater: \latex{12,150,301_6} or \latex{1,365,034_8}?
{{exercise_number}}. Convert the decimal number \latex{1,572} to the
- base-\latex{2} number system;
- base-\latex{4} number system;
- base-\latex{7} number system.
{{exercise_number}}. Convert the number \latex{34,251_6} to the base-\latex{4} number system.
{{exercise_number}}. What is the remainder if the number \latex{435,214_6} is divided by \latex{5}?
{{exercise_number}}. What is the remainder if the number \latex{23,012_4} is divided by \latex{5}?
{{exercise_number}}. With the help of the tables created in example \latex{4} do the following operations.
- \latex{3,421_{5}+{210,324_5}+{20,123_5}}
- \latex{4,203_{5}\times{324_5}}
- \latex{43,424_{5}+{34,443_5}}
- \latex{43,434_{5}\times{44,342_5}}
{{exercise_number}}. Objects with what mass can be measured in a balancing scale if a \latex{1} \latex{ kg }, a \latex{3} \latex{ kg }, a \latex{9} \latex{ kg } and a \latex{27} \latex{ kg } weight are available but we can place the weights into either pan?
Puzzle
Give numbers a, b, c so that \latex{3\times 6^{a}+2\times 6^{b}+6^{c}=23,100_{6}}.
