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Difference between revisions of "Base numbers"
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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 <math>2746</math>.  This number can be rewritten as <math>2746=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.  Also, each place in a number represents a value 10 times the place value to its right. 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. our example above may be written <math>2746_{10}</math>.
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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>
  
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What if we want to know how much <math>10010_2</math> is in base-10? (Notice how base 2 only uses digits 0 and 1.  For this reason, binary has many applications where only binary decisions, such as yes or no, are made, such as in computers.) From above, we see that each place value in binary is twice that of the place to its right. Hence, <math>10011_2=(1*2^4+0*2^3+0*2^2+1*2^1+1*2^0)_10=(16+2+1)_{10}=19_{10}</math>.
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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.
  
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Commonly used bases are 2, 8, 10 (duh!) and 16. The base doesn't necesarily have to be an integer. There are complex, irrational, negative, and many other kinds of bases. The best known one is [[phinary]], which is base [[phi]].
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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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== Common bases ==
 
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=== Binary ===
 
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Binary is base 2.  It's a favorite among computer programmers.
 
  
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=== Decimal ===
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== Base Number Topics ==
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Decimal is base 10.  It's the base that everyone knows and loves.  The most commonly used explanation for the origin of base 10 for our number system is the number of fingers we have.
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* [[base numbers/Common bases | Common bases]]
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* [[base numbers/Conversion | Converting between bases]]
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* [[Improper fractional base]]
  
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=== Hexadecimal ===
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== History ==
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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.
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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.
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== Example Problems ==
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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.
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Find the base 2 number that is equivalent to <math>42_7</math>
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=== Intermediate ===
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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]
  
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== History ==
 
  
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Base-10 is an apparently obvious counting system because people have 10 fingers. Historically, different societies utilized other systems. The Babylonians and some Native American cultures (Incas?) are known to have used base-60; this is why we say there are 360 degrees in a circle and (fact check on this coming one) 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, but rather 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 IIMD!  That's why the introduction of the '''Arab numberal system''', base-10, revolutionized math and science in Europe.
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== See Also ==
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*[[Number theory]]
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*[[Modular arithmetic]]
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*[http://www.artofproblemsolving.com/Forum/weblog_entry.php?t=92951 Richard Rusczyk's Base Number Article]

Latest revision as of 16:23, 30 December 2020

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 $42_7$

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

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