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.
The General Form
A number N written in base b with digits dn, dn-1, …, d1, d0 has the polynomial representation:
N = dn×bⁿ + dn-1×bn-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 × baseposition) 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.
Quick Practice
- Write 4,093 as a polynomial in base 10 and evaluate it.
- Evaluate 2×2³ + 1×2² + 0×2 + 1 in decimal.
- Write octal 52 as a polynomial in base 8 and find its decimal value.
- Explain why b⁰ = 1 means the rightmost digit always contributes its face value.
- Use the polynomial method to convert hex C4 to decimal.
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.