Polynomials/Numbers - Revision history

Admin at 21:52, 9 April 2017

2017-04-09T21:52:13Z

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 All of the real numbers of a given radix can also be expressed as a polynomial, where the variable is the radix.
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 \times 16^3 + 14 \times 16^2 + 14 \times 16^1 + 15 \times 16^0 = 48,879
 </math>
 {{PolynomialsFlag}}

Admin at 21:49, 9 April 2017

2017-04-09T21:49:08Z

← Older revision		Revision as of 21:49, 9 April 2017
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	</math>		</math>

			{{PolynomialsFlag}}
	~~[[Category:Polynomials]]~~
	~~[[Category:Math]]~~

Admin at 19:31, 9 April 2017

2017-04-09T19:31:38Z

← Older revision		Revision as of 19:31, 9 April 2017
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	<math>		<math>
	n = 125 = 1 \times 10^2 + 2 \times 10^1 + 5 \times 10^0		n = 125 = 1 \times 10^2 + 2 \times 10^1 + 5 \times 10^0
			</math>

			and if we replace the 10 with an x, we get a polynomial:

			<math>
			n = 125 = x^2 + 2 x + 5
	</math>		</math>

Admin: Created page with "All of the real numbers of a given radix can also be expressed as a polynomial, where the variable is the radix. For example, in base 10, we can split a number up into its on..."

2017-04-09T19:30:39Z

Created page with "All of the real numbers of a given radix can also be expressed as a polynomial, where the variable is the radix. For example, in base 10, we can split a number up into its on..."

New page

All of the real numbers of a given radix can also be expressed as a polynomial, where the variable is the radix.

For example, in base 10, we can split a number up into its ones, tens, hundreds, and so on. Suppose we have a number n,

<math>
n = 125 = 1 \times 100 + 2 \times 10 + 5 \times 1
</math>

This is equivalent to a polynomial representation in terms of the radix (10):

<math>
n = 125 = 1 \times 10^2 + 2 \times 10^1 + 5 \times 10^0
</math>

This generalizes to larger numbers:

<math>
n = 3,654,987 = 3 \times 10^6 + 6 \times 10^5 + 5 \times 10^4 + 4 \times 10^3 + 9 \times 10^2 + 8 \times 10^1 + 7 \times 10^0
</math>

That's just a polynomial with as many terms as digits, and the unknown <math>x</math> replacing the base:

<math>
n = 3 x^6 + 6 x^5 + 5 x^4 + 4 x^3 + 9 x^2 + 8 x + 7
</math>

It also generalizes to binary numbers or hex numbers:

<math>
n_{2} = 101010 = 1 \times 2^5 + 1 \times 2^3 + 1 \times 2^1 = 42
</math>

This number becomes the polynomial:

<math>
n_{2} = x^5 + x^3 + x
</math>

(where, for binary numbers, the set of possible coefficients is just the set <math>{0,1}</math>.)

Here is a hex number, converted to the equivalent polynomial:

<math>
n_{16} = BEEF = B \times 16^3 + E \times 16^2 + E \times 16^1 + F \times 16^0
</math>

Replacing the letters with their decimal equivalents, we get the base 10 equivalent number:

<math>
11 \times 16^3 + 14 \times 16^2 + 14 \times 16^1 + 15 \times 16^0 = 48,879
</math>

[[Category:Polynomials]]
[[Category:Math]]