Ostukorv on tühi
The binary numeral system
\latex{16} pence
\latex{8} pence
\latex{4} pence
\latex{2} pence
\latex{1} pence
The banknotes you see above are used in Binaria. Each banknote is worth as many pence as the number of dots on the given banknote. Pay \latex{ 13 } pence using the fewest banknotes possible.
organise the \latex{1}-pence
banknotes in pairs
banknotes in pairs
exchange them for
\latex{2}-pence banknotes
\latex{2}-pence banknotes
organise the \latex{2}-pence
banknotes in pairs
banknotes in pairs
exchange them for
\latex{4}-pence banknotes
\latex{4}-pence banknotes
organise the \latex{4}-pence
banknotes in pairs
banknotes in pairs
exchange them for
\latex{8}-pence banknotes
\latex{8}-pence banknotes
Write the number of banknotes you need in a table.
\latex{8}-pence banknotes
\latex{4}-pence banknotes
\latex{2}-pence banknotes
\latex{1}-pence banknotes
Total
paid
paid
\latex{1}
\latex{1}
\latex{0}
\latex{1}
\latex{13} pence
In the binary number system, numeric values are represented using only two digits. According to the previous example,
\latex{{\LARGE13_\text{\textcircled{\normalsize10}} \LARGE= 1101_\text{\textcircled{\normalsize2}}},}
where the number in the circle shows the number base of the system, indicating the number of digits used to represent numeric values. Unless you use the decimal number system, you must always indicate which number system you use.
\latex{ 1101 }\latex{_\text{\textcircled{2}}} should be read as one-one-zero-one in the binary numeral system.
The place value table of the binary numeral system
In the binary numeral system, numbers can be written in a base-\latex{ 2 } place value table using two digits (\latex{ 0 } and \latex{ 1 }).
\latex{\times }\latex{2}
\latex{\times }\latex{2}
\latex{\times }\latex{2}
\latex{\times }\latex{2}
\latex{\times }\latex{2}
...
thirty-twos
sixteens
eights
fours
twos
ones
...
\latex{1}
\latex{0}
\latex{1}
\latex{1}
\latex{0}
\latex{0}
\latex{1}
\latex{1}
\latex{1}
\latex{0}
\latex{1}
\latex{1}
\latex{1}
\latex{101}\latex{_\text{\textcircled{}}}
\latex{=}
\latex{1 \times 4 + 0 \times 2 + 1 \times 1 = 5}
2
\latex{1011}\latex{_\text{\textcircled{}}}
\latex{=}
2
\latex{1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1 = 11}
\latex{101011}\latex{_\text{\textcircled{}}}
2
\latex{=}
\latex{1 \times 32 + 0 \times 16 + 1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1 = 43}
Why are the real values of the digit \latex{ 1 } different in the number \latex{ 1101 }\latex{_\text{\textcircled{2}}}?
Digits
place value
face value
real value
\latex{1}
\latex{1}
\latex{0}
\latex{1}
eights
\latex{1}
\latex{8}
fours
\latex{1}
\latex{4}
twos
\latex{0}
\latex{0}
ones
\latex{1}
\latex{1}
place
value:
value:
fours
face
value:
value:
\latex{1}
real
value:
value:
\latex{4}
\latex{1} \latex{1} \latex{0} \latex{1}
Numbers in the binary numeral system in order are:
\latex{ 1 }; \latex{ 10 }; \latex{ 11 }; \latex{ 100 }; \latex{ 101 }; \latex{ 110 }; \latex{ 111 }; \latex{ 1000 }; \latex{ 1001 }; \latex{ 1010 }; ...
The following examples will show you how numbers can be converted from the binary numeral system to the decimal system and vice versa.
Example 1
Convert \latex{ 10111 }\latex{_\text{\textcircled{2}}} to the decimal system.
Solution
Write the number in the base-\latex{ 2 } place value table.
sixteens
eights
fours
twos
ones
\latex{1}
\latex{0}
\latex{1}
\latex{1}
\latex{1}
Based on this, the value of the number is:
\latex{ 1 \times 16 + 0 \times 8 + 1 \times 4 + 1 \times 2 + 1 \times 1 = 23.}
\latex{10111}\latex{_\text{\textcircled{2}}} as a decimal number is \latex{23}.
\latex{13}
\latex{\LARGE\circ}
10
\latex{1}
\latex{2}
\latex{4}
\latex{8}
\latex{16}
\latex{1}
\latex{4}
\latex{8}
Example 2
Convert the number \latex{ 27 } from the decimal to the binary system.
Solution
Use the place value table of the binary system.
\latex{27} \latex{\lt} \latex{32},
so no digits should be written in the thirty-two place value.
\latex{27} \latex{\gt} \latex{16},
thus there is
\latex{1} sixteens (and \latex{27} \latex{-} \latex{16} \latex{=} \latex{11} left).
\latex{11} \latex{\gt} \latex{8},
thus there is
\latex{1} eights (and \latex{11} \latex{-} \latex{8} \latex{=} \latex{3} left).
\latex{3} \latex{\lt} \latex{4},
thus there are
\latex{0} fours.
\latex{3} \latex{\gt} \latex{2},
thus there is
\latex{1} twos (and \latex{3} \latex{-} \latex{2} \latex{=} \latex{1} left).
\latex{1} is left, there is \latex{1} ones.
Write these in the place value table.
The
Decimal
System
Decimal
System
The
Binary
System
Binary
System
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{5}
\latex{6}
\latex{7}
\latex{8}
\latex{9}
\latex{10}
\latex{1010}\latex{_\text{\textcircled{2}}}
\latex{1001}\latex{_\text{\textcircled{2}}}
\latex{1000}\latex{_\text{\textcircled{2}}}
\latex{111}\latex{_\text{\textcircled{2}}}
\latex{110}\latex{_\text{\textcircled{2}}}
\latex{101}\latex{_\text{\textcircled{2}}}
\latex{100}\latex{_\text{\textcircled{2}}}
\latex{11}\latex{_\text{\textcircled{2}}}
\latex{10}\latex{_\text{\textcircled{2}}}
\latex{1}\latex{_\text{\textcircled{2}}}
sixteens
eights
fours
twos
ones
\latex{1}
\latex{1}
\latex{0}
\latex{1}
\latex{1}
thirty-twos
So \latex{ 27 }\latex{_\text{\textcircled{10}}} in the binary system is \latex{ 11011 }\latex{_\text{\textcircled{2}}} .
Everyday uses of the binary numeral system
Since only two symbols are used in the binary system, it is ideal for storing and sending information. Various devices, such as computers and digital cameras, process information as a series of signals consisting only of \latex{ 0 }s and \latex{ 1 }s. CDs, DVDs, and memory cards also store text, images, and audio files in this form.
Exercises
{{exercise_number}}. Which number do the following series of light bulbs correspond to in the binary system if every bulb turned on represents the \latex{ 1 }s, while every bulb turned off represents the \latex{ 0 }s?
a)
b)
c)
{{exercise_number}}. Convert the following numbers to the decimal system.
a) \latex{ 1011 }\latex{_\text{\textcircled{2}}}
b) \latex{ 1100 }\latex{_\text{\textcircled{2}}}
c) \latex{ 10001 }\latex{_\text{\textcircled{2}}}
d) \latex{ 11111 }\latex{_\text{\textcircled{2}}}
How can you tell at a glance whether a number in the binary system is even or not?
{{exercise_number}}. Convert the following numbers from the decimal to the binary system.
a) \latex{ 35 }
b) \latex{ 47 }
c) \latex{ 65 }
d) \latex{ 78 }
e) \latex{ 128 }
f) \latex{ 160 }
g) \latex{ 216 }
h) \latex{ 333 }
i) \latex{ 512 }
j) \latex{ 1,024 }
k) \latex{ 1,025 }
l) \latex{ 10,000 }
{{exercise_number}}. Write down ten numbers in the binary system,
a) counting by units;
b) counting by twos;
c) counting by fours.
{{exercise_number}}. Count backwards by twos from the following numbers. Write down the first five numbers in the binary and the decimal systems.
a) \latex{ 10110 }\latex{_\text{\textcircled{2}}}
b) \latex{ 10010 }\latex{_\text{\textcircled{2}}}
c) \latex{ 10001 }\latex{_\text{\textcircled{2}}}
d) \latex{ 10101 }\latex{_\text{\textcircled{2}}}
e) \latex{ 10000 }\latex{_\text{\textcircled{2}}}
{{exercise_number}}. How many whole numbers in the binary system consist of
a) one digit;
b) two digits;
c) three digits;
d) four digits?
Quiz
How many two-digit numbers are there in the base-\latex{ 5 } numeral system?
