Unit 1: Speed Maths, Advanced Numeration and Mean

1. Algorithmic Speed Mathematics and Mental Computation

1.1 Foundational Sutras of Vedic Mathematics

Speed mathematics leverages structural properties of positional base-10 numeration to transform polynomial-time arithmetic operations into linear-time mental shortcuts.

1. Ekadhikena Purvena ("By One More Than the Previous"):

   • Squaring Numbers Ending in 5:
     Any positive integer ending in 5 can be represented as $N = 10a + 5$, where $a \in \mathbb{N}_0$.
     $N^2 = (10a + 5)^2 = 100a^2 + 100a + 25 = 100a(a + 1) + 25$
     The left part of the square is computed by multiplying the prefix $a$ by its successor $(a + 1)$, while the terminal two digits are invariably $25$.
     Example: $85^2 \implies a = 8$. Left part: $8 \times 9 = 72$; Right part: $25 \implies 7225$.
   • Recurring Decimal Expansions (Denominators ending in 9):
     Computing the decimal expansion of $\frac{1}{19}$ using the auxiliary fraction $\frac{1}{20}$: beginning from the last digit, each preceding digit is obtained by multiplying by 2 (or dividing sequentially).

2. Nikhilam Navatashcaramam Dashatah ("All from 9 and the Last from 10"):

   • Multiplication Near a Common Power-of-10 Base ($B = 10^k$):
     Let two numbers be $X = B + d_1$ and $Y = B + d_2$, where $d_1, d_2$ represent positive or negative base deviations.
     $X \cdot Y = (B + d_1)(B + d_2) = B(B + d_1 + d_2) + d_1 d_2 = B(X + d_2) + d_1 d_2$
     Example (Deficit Base 1000): $994 \times 988$. Base $B = 1000$. Deviations $d_1 = -6, d_2 = -12$.
     • Left part: $X + d_2 = 994 + (-12) = 982$.
     • Right part: $d_1 \times d_2 = (-6) \times (-12) = +72$.
       Since base $1000$ requires 3 digits on the right: $072$.
       Product $= 982,072$.
   • Surplus-Deficit Hybrid Multiplication:
     When $X = B + d_1$ (surplus) and $Y = B - d_2$ (deficit):
     $X \cdot Y = B(X - d_2) - d_1 d_2 = [B(X - d_2) - 1] + [B - d_1 d_2]$

3. Urdhva Tiryagbhyam ("Vertically and Crosswise"):
   General multiplication algorithm for arbitrary multi-digit numbers, mathematically equivalent to discrete polynomial convolution.
   For two 2-digit numbers $(10a + b)$ and $(10c + d)$:
   $(10a + b)(10c + d) = 100(ac) + 10(ad + bc) + (bd)$

   • Step 1: Multiply vertical right column: $b \times d$. Record unit digit, carry tens.
   • Step 2: Multiply crosswise and sum: $(a \times d) + (b \times c) + \text{carry}$.
   • Step 3: Multiply vertical left column: $(a \times c) + \text{carry}$.

   For 3-digit multiplication $(100a + 10b + c) \times (100d + 10e + f)$:

   • Column 1: $c \times f$
   • Column 2: $b \times f + c \times e$
   • Column 3: $a \times f + b \times e + c \times d$
   • Column 4: $a \times e + b \times d$
   • Column 5: $a \times d$

4. Vinculum Representation (Signed-Digit Numeration):
   A method to replace digits greater than 5 with negative digits ($\bar{d}$ denoting $-d$), minimizing mental carry operations:
   $10a + b = 10(a + 1) - (10 - b) = 10(a + 1) + \overline{10 - b}$
   Example: $389 = 400 - 11 = 4\bar{1}\bar{1}$.
   Squaring $389$: $(400 - 11)^2 = 160000 - 8800 + 121 = 151,321$.

1.2 Fast Duplex Extraction of Roots (Dvandva Yoga)

The Duplex ($D$) of a sequence of digits is defined:

• For a single digit $a$: $D(a) = a^2$.
• For two digits $ab$: $D(ab) = 2ab$.
• For three digits $abc$: $D(abc) = 2ac + b^2$.
• For four digits $abcd$: $D(abcd) = 2ad + 2bc$.
• For $n$ digits $d_1 d_2 \dots d_n$: $D = 2(d_1 d_n + d_2 d_{n-1} + \dots) + d_{(n+1)/2}^2$ (if $n$ is odd).

Duplex extraction enables mental determination of square roots by subtracting successive duplexes from running remainders.

2. Advanced Numeration and Number Theory

2.1 The Fundamental Theorem of Arithmetic and Divisor Algebra

Every integer $N > 1$ admits a unique canonical prime factorization up to the order of factors:
$N = p_1^{a_1} p_2^{a_2} p_3^{a_3} \cdots p_k^{a_k} = \prod_{i=1}^k p_i^{a_i}$
where $p_i$ are distinct prime numbers and $a_i \in \mathbb{N}$.

1. Total Number of Divisors $d(N)$ or $\tau(N)$:

Every divisor of $N$ takes the form $d = p_1^{b_1} p_2^{b_2} \cdots p_k^{b_k}$ with $0 \le b_i \le a_i$. By the rule of product:
$d(N) = (a_1 + 1)(a_2 + 1)(a_3 + 1)\cdots(a_k + 1) = \prod_{i=1}^k (a_i + 1)$

2. Sum of All Positive Divisors $\sigma(N)$:

Expanding the geometric series across each prime component:
$\sigma(N) = \left( \sum_{b_1=0}^{a_1} p_1^{b_1} \right) \left( \sum_{b_2=0}^{a_2} p_2^{b_2} \right) \cdots \left( \sum_{b_k=0}^{a_k} p_k^{b_k} \right) = \prod_{i=1}^k \frac{p_i^{a_i + 1} - 1}{p_i - 1}$

3. Product of All Divisors $P(N)$:

Pairing each divisor $d$ with its complementary cofactor $N / d$:
$P(N) = N^{\frac{d(N)}{2}}$

4. Number of Ways to Factor $N$ as a Product of Two Factors:

$\text{Ways} = \begin{cases} \frac{d(N)}{2} & \text{if } N \text{ is not a perfect square} \\ \frac{d(N) + 1}{2} & \text{if } N \text{ is a perfect square (distinct or identical)} \end{cases}$

5. Even and Odd Divisors:

Let $N = 2^a \cdot p_2^{a_2} \cdots p_k^{a_k}$ with $a \ge 1$:

• Total Odd Divisors: $d_{\text{odd}}(N) = (a_2 + 1)(a_3 + 1)\cdots(a_k + 1)$.
• Total Even Divisors: $d_{\text{even}}(N) = d(N) - d_{\text{odd}}(N) = a(a_2 + 1)\cdots(a_k + 1)$.

2.2 Euler's Totient Function $\phi(N)$ and Modular Theorems

Euler's totient function counts the number of positive integers up to $N$ that are coprime to $N$ ($\gcd(k, N) = 1$ for $1 \le k \le N$):
$\phi(N) = N \prod_{p \mid N} \left( 1 - \frac{1}{p} \right) = N \left( 1 - \frac{1}{p_1} \right)\left( 1 - \frac{1}{p_2} \right)\cdots\left( 1 - \frac{1}{p_k} \right)$

• If $p$ is prime: $\phi(p) = p - 1$.
• If $p$ is prime and $a \ge 1$: $\phi(p^a) = p^a - p^{a-1} = p^a \left(1 - \frac{1}{p}\right)$.
• Multiplicative Property: If $\gcd(m, n) = 1$, then $\phi(mn) = \phi(m)\phi(n)$.

1. Euler's Totient Theorem:

If $a$ and $m$ are coprime positive integers ($\gcd(a, m) = 1$):
$a^{\phi(m)} \equiv 1 \pmod m$

2. Fermat's Little Theorem:

If $p$ is a prime and $\gcd(a, p) = 1$:
$a^{p - 1} \equiv 1 \pmod p \implies a^p \equiv a \pmod p$

3. Wilson's Theorem:

A natural number $p > 1$ is prime if and only if:
$(p - 1)! \equiv -1 \equiv p - 1 \pmod p$

2.3 Cyclicity, Residue Arithmetic, and Units Digit Determination

Powers of single digits exhibit cyclic periodicity in their unit digits modulo 10:

• Cyclicity 1: $\{0, 1, 5, 6\}$ (terminal digit is invariant for all powers $\ge 1$).
• Cyclicity 2: $\{4, 9\}$ ($4^{\text{odd}} \equiv 4, 4^{\text{even}} \equiv 6$; $9^{\text{odd}} \equiv 9, 9^{\text{even}} \equiv 1$).
• Cyclicity 4: $\{2, 3, 7, 8\}$:
  • $2^1 \equiv 2, 2^2 \equiv 4, 2^3 \equiv 8, 2^4 \equiv 6 \pmod{10}$
  • $3^1 \equiv 3, 3^2 \equiv 9, 3^3 \equiv 7, 3^4 \equiv 1 \pmod{10}$
  • $7^1 \equiv 7, 7^2 \equiv 9, 7^3 \equiv 3, 7^4 \equiv 1 \pmod{10}$
  • $8^1 \equiv 8, 8^2 \equiv 4, 8^3 \equiv 2, 8^4 \equiv 6 \pmod{10}$

To compute the units digit of $A^B$:

1. Reduce base $A$ modulo 10: $a = A \pmod{10}$.
2. Compute $r = B \pmod 4$. If $r = 0$, set exponent power to 4.
3. Compute $a^r \pmod{10}$.

Terminal Two Digits (Modulo 100):

To compute the last two digits of $A^B$:

1. If $A$ ends in 1: $(10a + 1)^B \equiv 10(a \cdot B) + 1 \pmod{100}$.
2. If $A$ ends in 5: $A^2 \equiv 25 \pmod{100}$.
3. If $A$ is even: Factor out $2^k$, utilize $2^{10} = 1024 \equiv 24 \pmod{100}$, and $24^{\text{even}} \equiv 76 \pmod{100}$.

2.4 Legendre's Formula (Highest Power of Prime Dividing $n!$)

The exponent of a prime $p$ in the prime factorization of $n!$, denoted $E_p(n!)$ or $v_p(n!)$, is given by Legendre's de Polignac formula:
$E_p(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \dots$

Trailing Zeros in $n!$:

A trailing zero is generated by the prime factor $10 = 2 \times 5$. In any factorial $n!$, the factor 2 is far more abundant than 5 ($E_2(n!) > E_5(n!)$). Therefore, the number of trailing zeros $Z(n!)$ is strictly governed by $E_5(n!)$:
$Z(n!) = E_5(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{5^k} \right\rfloor = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \dots$

3. The Chinese Remainder Theorem (CRT) and Modular Congruences

3.1 Constructive Theorem Formulation

Let $m_1, m_2, \dots, m_k$ be pairwise coprime positive integers ($\gcd(m_i, m_j) = 1$ for $i \ne j$), and let $a_1, a_2, \dots, a_k$ be arbitrary integers.
The system of simultaneous linear congruences:
$x \equiv a_1 \pmod{m_1}$
$x \equiv a_2 \pmod{m_2}$
$\vdots$
$x \equiv a_k \pmod{m_k}$
admits a unique solution modulo $M = \prod_{i=1}^k m_i$.

3.2 Constructive Solution Algorithm

1. Compute the global modulus: $M = m_1 \cdot m_2 \cdots m_k$.
2. For each index $i \in \{1, \dots, k\}$, compute the cofactor:
   $M_i = \frac{M}{m_i} = \prod_{j \ne i} m_j$
3. By coprimality $\gcd(M_i, m_i) = 1$, find the modular multiplicative inverse $y_i = M_i^{-1} \pmod{m_i}$ using the Extended Euclidean Algorithm such that:
   $M_i \cdot y_i \equiv 1 \pmod{m_i}$
4. The unique minimal non-negative solution is:
   $x = \left( \sum_{i=1}^k a_i M_i y_i \right) \pmod M$

4. Positional Radix Numeration and Cross-Base Conversions

In computer engineering and discrete analytics, numbers are represented across distinct radix bases $b \ge 2$:
$N = (d_k d_{k-1} \dots d_1 d_0)_b = \sum_{i=0}^k d_i \cdot b^i, \quad 0 \le d_i < b$

4.1 Radix Conversion Mechanisms

1. Base $b \to$ Decimal (Base 10): Direct polynomial evaluation via Horner's rule:
   $N = (\dots((d_k \cdot b + d_{k-1}) \cdot b + d_{k-2})\dots) \cdot b + d_0$
2. Decimal $\to$ Base $b$: Successive integer division by $b$, recording remainders from least significant digit ($d_0$) to most significant digit ($d_k$).
3. Power-of-2 Grouping (Binary, Octal, Hexadecimal):
   • Base 2 to Base 8: Group bits into clusters of 3 ($2^3 = 8$).
   • Base 2 to Base 16: Group bits into clusters of 4 ($2^4 = 16$).

5. Advanced Theory of Means and Statistical Centrality

5.1 The Classical Pythagorean Means

For a set of $n$ strictly positive real numbers $\{x_1, x_2, \dots, x_n\} \subset \mathbb{R}^+$:

1. Arithmetic Mean ($AM$):
   The balance point or first central moment:
   $AM = \frac{1}{n} \sum_{i=1}^n x_i = \frac{x_1 + x_2 + \dots + x_n}{n}$

2. Geometric Mean ($GM$):
   The multiplicative central measure, invariant under proportional compounding:
   $GM = \left( \prod_{i=1}^n x_i \right)^{1/n} = \sqrt[n]{x_1 \cdot x_2 \cdots x_n}$
   In logarithmic form:
   $\ln(GM) = \frac{1}{n} \sum_{i=1}^n \ln(x_i)$

3. Harmonic Mean ($HM$):
   The reciprocal of the arithmetic mean of reciprocals, governing rates and ratios:
   $HM = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}}$

4. Root Mean Square / Quadratic Mean ($RMS$ or $QM$):
   Governing alternating waveforms, statistical variance, and error metrics:
   $RMS = \sqrt{\frac{1}{n} \sum_{i=1}^n x_i^2} = \sqrt{\frac{x_1^2 + x_2^2 + \dots + x_n^2}{n}}$

5. Contraharmonic Mean ($CHM$):
   Dual to the Harmonic Mean, placing greater weight on larger values:
   $CHM = \frac{\sum_{i=1}^n x_i^2}{\sum_{i=1}^n x_i} = \frac{x_1^2 + x_2^2 + \dots + x_n^2}{x_1 + x_2 + \dots + x_n}$
   Notice that $CHM + HM = 2 \cdot AM$ for two numbers.

5.2 The Generalized Hölder Power Mean ($M_p$)

Pythagorean and quadratic means are unified under the one-parameter Hölder Power Mean family:
$M_p(x_1, \dots, x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p}, \quad p \in \mathbb{R} \setminus \{0\}$

• $p \to -\infty \implies \min(x_1, \dots, x_n)$
• $p = -1 \implies HM$
• $p \to 0 \implies GM$ (via L'Hôpital's rule on logarithmic limits)
• $p = 1 \implies AM$
• $p = 2 \implies RMS$
• $p \to +\infty \implies \max(x_1, \dots, x_n)$

5.3 The Fundamental Mean Inequality

For any collection of positive real numbers $x_1, x_2, \dots, x_n > 0$:
$\min(x_i) \le HM \le GM \le AM \le RMS \le CHM \le \max(x_i)$
Equality holds throughout if and only if all variables are strictly identical: $x_1 = x_2 = \dots = x_n$.

CODECopy Code
              The Hierarchy of Centrality Measures
  +-----------------------------------------------------------------------+
  | Min  <=  HM  <=  GM  <=  AM  <=  RMS  <=  Contraharmonic  <=  Max     |
  +-----------------------------------------------------------------------+

For two positive numbers $a$ and $b$:
$AM = \frac{a + b}{2}, \quad GM = \sqrt{ab}, \quad HM = \frac{2ab}{a + b}$
$GM^2 = AM \cdot HM \implies GM = \sqrt{AM \cdot HM}$
Furthermore:
$AM - GM = \frac{a + b - 2\sqrt{ab}}{2} = \frac{(\sqrt{a} - \sqrt{b})^2}{2} \ge 0$
which establishes an analytical proof of $AM \ge GM$.

5.4 Rate Averaging and Harmonic Mean Applications

When computing the average rate across multiple stages:

1. Equal Distances, Variable Speeds:
   Let a vehicle travel distance $D$ at speed $v_1$, and the identical distance $D$ at speed $v_2$.
   $\text{Total Time } T = \frac{D}{v_1} + \frac{D}{v_2} = D \left( \frac{v_1 + v_2}{v_1 v_2} \right)$
   $\text{Average Speed } v_{\text{avg}} = \frac{2D}{T} = \frac{2D}{D \left( \frac{v_1 + v_2}{v_1 v_2} \right)} = \frac{2 v_1 v_2}{v_1 + v_2} = HM(v_1, v_2)$
2. Equal Times, Variable Speeds:
   If a vehicle travels for time $t$ at speed $v_1$, and for the identical time $t$ at speed $v_2$:
   $\text{Total Distance } D = v_1 t + v_2 t = (v_1 + v_2) t$
   $\text{Average Speed } v_{\text{avg}} = \frac{D}{2t} = \frac{(v_1 + v_2) t}{2t} = \frac{v_1 + v_2}{2} = AM(v_1, v_2)$

5.5 Robust Means: Trimmed and Winsorized Estimators

When data sets contain extreme outliers or heavy-tailed noise:

1. Trimmed Mean ($\alpha$-trimmed): Discards the lowest and highest $k = \lfloor \alpha \cdot n \rfloor$ observations before computing the arithmetic mean:
   $\bar{x}_{\text{trim}} = \frac{1}{n - 2k} \sum_{i=k+1}^{n-k} x_{(i)}$
2. Winsorized Mean: Replaces the lowest $k$ values with $x_{(k+1)}$ and highest $k$ values with $x_{(n-k)}$, preserving sample size $n$ while suppressing outlier influence.

6. Comprehensive Mathematical Reference Formulary