Difference between revisions of "Base numbers"

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(Base Number Topics)
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To understand the notion of base numbers, we look at our own number system.  We use the '''decimal''', or base-10, number system.  To help explain what this means, consider the number 2746.  This number can be rewritten as <center><math>\displaystyle 2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math></center>
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To understand the notion of '''base numbers''', we look at our own [[number system]].  We use the [[decimal]], or base-10, number system.  To help explain what this means, consider the number 2746.  This number can be rewritten as <math>2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math>
  
Note that each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are.  The first digit to the left of the decimal place (recall that the decimal place is to the right of the 6, i.e. 2746.0) tells us that there are six <math>10^0</math>'s, the second digit tells us there are four <math>10^1</math>'s, the third digit tells us there are seven <math>10^2</math>'s, and the fourth digit tells us there are two <math>\displaystyle 10^3</math>'s.
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Note that each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are.  The first digit to the left of the decimal place (recall that the decimal place is to the right of the 6, i.e. 2746.0) tells us that there are six <math>10^0</math>'s, the second digit tells us there are four <math>10^1</math>'s, the third digit tells us there are seven <math>10^2</math>'s, and the fourth digit tells us there are two <math>10^3</math>'s.
  
 
Base-10 uses digits 0-9.  Usually, the base, or '''radix''', of a number is denoted as a subscript written at the right end of the number (e.g. in our example above, <math>2746_{10}</math>, 10 is the radix).
 
Base-10 uses digits 0-9.  Usually, the base, or '''radix''', of a number is denoted as a subscript written at the right end of the number (e.g. in our example above, <math>2746_{10}</math>, 10 is the radix).
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* [[base numbers/Common bases | Common bases]]
 
* [[base numbers/Common bases | Common bases]]
 
* [[base numbers/Conversion | Converting between bases]]
 
* [[base numbers/Conversion | Converting between bases]]
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* [[Improper fractional base]]
  
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== History ==
  
== Common bases ==
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Base-10 is an apparently obvious counting system because people have 10 fingers.  Historically, different societies utilized other systems.  The  Babylonian cultures are known to have used base-60; this is why we say there are 360 degrees in a circle and (fact check on this one coming) why we count 60 minutes in an hour and 60 seconds in a minute (they might have used it because it has so many multiples, 12 in fact, we wouldn't want any fractions). The [[Roman system]], which didn't have any base system at all, used certain letters to represent certain values (e.g. I=1, V=5, X=10, L=50, C=100, D=500, M=1000).  Imagine how difficult it would be to multiply LXV by MDII!  That's why the introduction of the '''Arabic numeral system''', base-10, revolutionized math and science in Europe.
Commonly used bases are 2, 8, 10 (duh!) and 16. The base doesn't necesarily have to be an integer. There are [[complex base | complex]], [[irrational base | irrational]], [[negative base | negative]], [[improper fractional base | fractional]], and many other kinds of bases. The best known one is [[phinary]], which is base [[phi]]; others include "Fibonacci base" and base negative two.
 
  
=== Binary ===
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== Example Problems ==
Binary is base 2.  It's a favorite among computer programmers. It has just two digits: <math>0</math> and <math>1</math>.
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=== Beginner ===
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*Evaluate <math>\sqrt{61_{8}}</math> as a number in the decimal system.
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**Solution: <math>61_{8}</math> must be rewritten in the decimal system (base-10) before evaluating the square root. To do this, multiply and add <math>6\cdot 8^1+1\cdot 8^0=48+1=49.  \sqrt{49}=7.</math> Therefore, the answer is 7.
  
===Octal ===
 
Octal is base 8. It was also quite liked by programmers because the octal representation of numbers is 3 times shorter than the binary one and the conversion from octal to binary and back is very easy (can you guess why?). Besides, 8 is quite close to 10 and less than 10, so to learn doing addition and multiplication in base 8 is not very hard: you can basically count in base 10 with partial conversions to base 8 on the way. Let's multiply <math>12345_8</math> by <math>7_8</math>. <math>5\cdot 7=35_{10}=43_8</math> (to get the last result, just divide <math>35</math> by <math>8</math> with remainder). As usual, we write  the last digit <math>3</math> down and keep <math>4</math> in mind. Now, <math>4\cdot 7+4=32_{10}=40_8</math>, so we write down <math>0</math>, getting <math>03</math>, and keeping <math>4</math> in mind.
 
And so on. The time needed to get the answer <math>111103_8</math> only marginally exceeds the time of decimal multiplication (if you are good in division by 8 with remainder, of course).
 
  
=== Decimal ===
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Find the base 2 number that is equivalent to <math>49_7</math>
Decimal is base 10.  It's the base that everyone knows and loves.  Most numbers in the world are written without a specified radix and  usually it can just be assumed that they are in base 10.  The most commonly used explanation for the origin of base 10 for our number system is the number of fingers we have.
 
  
=== Hexadecimal ===
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=== Intermediate ===
Hexadecimal is base 16.  The digits in hexadecimal are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.  One of its common uses is for color charts. Hexadecimal numbers are also used by programmers in the same way as octal numbers, but to learn to count in hexadecimal is harder than in octal.
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* [[2003_AIME_I_Problems/Problem_13 | 2003 AIME I Problem 13]]
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* [[1977_Canadian_MO_Problems/Problem_3 | Canadian Mathematics Olympiad Problem 3]]
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* Suppose <math>P(x)</math> is an unknown polynomial, of unknown degree, with nonnegative integer coefficients. Your goal is to determine this polynomial. You have access to an oracle that, given an integer <math>n</math>, spits out <math>P(n)</math>, the value of the polynomial at <math>n</math>. However, the oracle charges a fee for each such computation, so you want to minimize the number of computations you ask the oracle to do. Show that it is possible to uniquely determine the polynomial after only two consultations of the oracle. ([http://www.math.uiuc.edu/~hildebr/pow/pow10.pdf UIUC POW])
  
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== Resources ==
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==== Books ====
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* The AoPS [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=10 Introduction to Number Theory] by [[Mathew Crawford]].
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==== Classes ====
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* [http://www.artofproblemsolving.com/Classes/AoPS_C_ClassesS.php#begnum AoPS Introduction to Number Theory Course]
  
== History ==
 
 
Base-10 is an apparently obvious counting system because people have 10 fingers.  Historically, different societies utilized other systems.  The  Native American cultures are known to have used base-60; this is why we say there are 360 degrees in a circle and (fact check on this one coming) why we count 60 minutes in an hour and 60 seconds in a minute.  The Roman system (internal link w/explanation?), which didn't have any base system at all, used certain letters to represent certain values (e.g. I=1, V=5, X=10, L=50, C=100, D=500, M=1000).  Imagine how difficult it would be to multiply LXV by MDII!  That's why the introduction of the '''Arabic numeral system''', base-10, revolutionized math and science in Europe.
 
 
 
== Example Problems ==
 
=== Intermediate ===
 
* [[1977_Canadian_MO_Problems/Problem_3 1977 | Canadian Mathematics Olympiad Problem 3]]
 
  
 
== See Also ==
 
== See Also ==

Revision as of 11:14, 9 October 2017

To understand the notion of base numbers, we look at our own number system. We use the decimal, or base-10, number system. To help explain what this means, consider the number 2746. This number can be rewritten as $2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.$

Note that each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are. The first digit to the left of the decimal place (recall that the decimal place is to the right of the 6, i.e. 2746.0) tells us that there are six $10^0$'s, the second digit tells us there are four $10^1$'s, the third digit tells us there are seven $10^2$'s, and the fourth digit tells us there are two $10^3$'s.

Base-10 uses digits 0-9. Usually, the base, or radix, of a number is denoted as a subscript written at the right end of the number (e.g. in our example above, $2746_{10}$, 10 is the radix).


Base Number Topics

History

Base-10 is an apparently obvious counting system because people have 10 fingers. Historically, different societies utilized other systems. The Babylonian cultures are known to have used base-60; this is why we say there are 360 degrees in a circle and (fact check on this one coming) why we count 60 minutes in an hour and 60 seconds in a minute (they might have used it because it has so many multiples, 12 in fact, we wouldn't want any fractions). The Roman system, which didn't have any base system at all, used certain letters to represent certain values (e.g. I=1, V=5, X=10, L=50, C=100, D=500, M=1000). Imagine how difficult it would be to multiply LXV by MDII! That's why the introduction of the Arabic numeral system, base-10, revolutionized math and science in Europe.

Example Problems

Beginner

  • Evaluate $\sqrt{61_{8}}$ as a number in the decimal system.
    • Solution: $61_{8}$ must be rewritten in the decimal system (base-10) before evaluating the square root. To do this, multiply and add $6\cdot 8^1+1\cdot 8^0=48+1=49.  \sqrt{49}=7.$ Therefore, the answer is 7.


Find the base 2 number that is equivalent to $49_7$

Intermediate

  • 2003 AIME I Problem 13
  • Canadian Mathematics Olympiad Problem 3
  • Suppose $P(x)$ is an unknown polynomial, of unknown degree, with nonnegative integer coefficients. Your goal is to determine this polynomial. You have access to an oracle that, given an integer $n$, spits out $P(n)$, the value of the polynomial at $n$. However, the oracle charges a fee for each such computation, so you want to minimize the number of computations you ask the oracle to do. Show that it is possible to uniquely determine the polynomial after only two consultations of the oracle. (UIUC POW)

Resources

Books

Classes


See Also