Polynomial Representation

Polynomial representation is the formal mathematical way of expressing numbers in any base using algebraic notation. It shows exactly how place value works as a sum of terms, where each term is a digit multiplied by a power of the base.

Efficiently evaluating a sum of this kind has a long history. Methods for it date back to the 13th-century Chinese mathematician Qin Jiushao, and were rediscovered in the West by William Horner in 1819 — a technique now known as Horner's method. Rather than computing each power of the base separately, Horner's method works from the leftmost digit outward, repeatedly multiplying by the base and adding the next digit, using far fewer multiplications. This is still how calculators, computers, and base-conversion algorithms evaluate numbers internally today.

The General Form

A number N written in base b with digits d_n, d_n-1, …, d₁, d₀ has the polynomial representation:

N = d_n×bⁿ + d_n-1×b^n-1 + … + d₁×b¹ + d₀×b⁰

Base 10 (Decimal) Examples

Number	Polynomial (base 10)	Evaluation
83	8×10¹ + 3×10⁰	80 + 3 = 83
507	5×10² + 0×10¹ + 7×10⁰	500 + 0 + 7 = 507
2,064	2×10³ + 0×10² + 6×10¹ + 4×10⁰	2,000+0+60+4 = 2,064

Base 2 (Binary) Examples

Binary	Polynomial (base 2)	Decimal
101	1×2² + 0×2¹ + 1×2⁰	4+0+1 = 5
1100	1×2³ + 1×2² + 0 + 0	8+4 = 12
11111	16+8+4+2+1	31

Base 16 (Hex) Example

3B (hex) = 3×16¹ + 11×16⁰ = 48 + 11 = 59 (decimal)

Polynomial Addition – Carry Mechanism

When adding two numbers in polynomial form, if a coefficient reaches or exceeds the base, we carry 1 to the next higher power:

38 + 57 = (3+5)×10 + (8+7) = 8×10 + 15 → carry: (8+1)×10 + 5 = 95

Why This Matters

It proves that our place value system is a direct application of polynomial algebra.
It provides a unified view of all positional number systems.
It leads to Horner's method – an efficient way to evaluate polynomials.
It connects arithmetic to abstract algebra and number theory.

Key Points

Any positional number = sum of (digit × base^position) for each digit.
The rightmost digit has exponent 0 (base⁰ = 1).
The same formula works for any base: 2, 8, 10, 16, or any integer ≥2.
Carrying in arithmetic is the polynomial equivalent of coefficient reduction modulo the base.

Practice: Evaluate the Polynomial

A number is shown below in polynomial form, in base 2, 8, or 16. Enter its decimal value. If you get it wrong (or leave it blank), we'll evaluate each term together.

Evaluate this polynomial to find the decimal value.

Summary

Polynomial representation reveals that place value is not an arbitrary system – it is a polynomial in the base. This insight unifies arithmetic across all number bases, explains carrying, and connects elementary number work to the world of polynomial algebra and number theory.