BASIC ALGEBRA & ARITHMETIC

WHY THIS MATTERS

Algebra is the language of Physics. Every single physical equation — from Newton's second law ($F = ma$) to Einstein's mass–energy equivalence ($E = mc^2$) — is an algebraic relationship. Mastering manipulation of these equations is the very first skill a Physics student must develop.

§1. Fractions, Indices & Surds

Laws of Indices (Exponents)

When quantities are raised to powers, we use the following rules to simplify them:

Product Rule

$a^m \cdot a^n = a^{m+n}$

Ex: $a^3 \cdot a^5 = a^8$

Quotient Rule

$\frac{a^m}{a^n} = a^{m-n}$

Ex: $\frac{x^7}{x^3} = x^4$

Power of Power

$(a^m)^n = a^{mn}$

Ex: $(x^2)^3 = x^6$

Zero & Negative

$a^0 = 1 \quad a^{-n} = \frac{1}{a^n}$

Ex: $x^{-2} = \frac{1}{x^2}$

Fractional Exponents

$a^{1/n} = \sqrt[n]{a}$

Ex: $x^{1/2} = \sqrt{x}, \quad x^{3/2} = x\sqrt{x}$

Distributing Powers

$(ab)^n = a^n b^n$

$(\frac{a}{b})^n = \frac{a^n}{b^n}$

⚛️ Physics Use:

Gravitational force $F \propto r^{-2}$, Electric field $E \propto r^{-2}$ — these negative exponents come directly from index laws!

§2. Quadratic Equations

A quadratic equation has the standard form: $ax^2 + bx + c = 0$, where $a \neq 0$.

The Quadratic Formula (Sridharacharya Formula)

MASTER FORMULA

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The term $\Delta = b^2 - 4ac$ is called the Discriminant. It tells us the nature of roots:

If $\Delta > 0$: Two distinct real roots
If $\Delta = 0$: Two equal (repeated) real roots
If $\Delta < 0$: No real roots (imaginary roots — not relevant in standard physics)

Properties of Roots

If $\alpha$ and $\beta$ are the two roots of $ax^2 + bx + c = 0$, then:

Sum of Roots

$$\alpha + \beta = -\frac{b}{a}$$

Product of Roots

$$\alpha \cdot \beta = \frac{c}{a}$$

Solved Example: Find the roots of $2x^2 - 7x + 3 = 0$.

Here, $a = 2, b = -7, c = 3$.
$\Delta = b^2 - 4ac = 49 - 24 = 25$
$x = \frac{7 \pm \sqrt{25}}{4} = \frac{7 \pm 5}{4}$
$\therefore x = \frac{12}{4} = 3 \quad \text{or} \quad x = \frac{2}{4} = \frac{1}{2}$

⚛️ Physics Use:

When solving projectile motion problems for time of flight (the ball returns to ground), you get a quadratic equation in time $t$: $h = ut - \frac{1}{2}gt^2$, giving $t$ via the quadratic formula.

§3. Binomial Theorem & Approximations

Graph: Comparing exact $(1 + x)^2$ vs. approximation $1 + 2x$ for small values of x

The Binomial Theorem gives the expansion of $(1 + x)^n$ for any real value of $n$:

BINOMIAL EXPANSION

$$(1 + x)^n = 1 + nx + \frac{n(n - 1)}{2!}x^2 + \frac{n(n - 1)(n - 2)}{3!}x^3 + \dots$$

The Physics Approximation (Most Important!)

⭐ CRITICAL FORMULA

$$\text{If } |x| \ll 1, \text{ then } (1 + x)^n \approx 1 + nx$$

All higher-power terms of $x$ become negligibly small and can be dropped.

Solved Example: Simplify $\sqrt{1 + 0.02}$ approximately.

Write as $(1 + 0.02)^{1/2}$. Here $n = 1/2$ and $x = 0.02$ (which is $\ll 1$).
$\approx 1 + \frac{1}{2}(0.02) = 1 + 0.01 = \mathbf{1.01}$
(Exact value: 1.00995 — the approximation is extremely accurate!)

⚛️ Physics Uses:

Gravitation: $g' = g\left(1 - \frac{2h}{R}\right)$ — derived using binomial approx on $\left(\frac{R}{R+h}\right)^2 \approx 1 - \frac{2h}{R}$
Error Analysis: $\frac{\Delta Z}{Z} = n \frac{\Delta A}{A}$ comes from $(A + \Delta A)^n \approx A^n\left(1 + n\frac{\Delta A}{A}\right)$

§4. Arithmetic & Geometric Progressions

Arithmetic Progression (AP)

A sequence where each term differs by a constant amount (common difference, $d$).
General form: $a, \ a + d, \ a + 2d, \ a + 3d, \dots$

$n^{th}$ Term

$$a_n = a + (n - 1)d$$

Sum of $n$ terms

$$S_n = \frac{n}{2}[2a + (n - 1)d]$$ $$\text{or } S_n = \frac{n}{2}(a + l)$$

Geometric Progression (GP)

A sequence where each term is obtained by multiplying the previous by a constant (common ratio, $r$).
General form: $a, \ ar, \ ar^2, \ ar^3, \dots$

$n^{th}$ Term

$$a_n = ar^{n-1}$$

Sum of $n$ terms ($r \neq 1$)

$$S_n = \frac{a(1 - r^n)}{1 - r}$$

Sum of Infinite GP

(only if $|r| < 1$)

$$S_\infty = \frac{a}{1 - r}$$

Solved Example (Infinite GP): Find $1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots$

Here $a = 1$, $r = 1/3$ (which is $|r| < 1$).
$S_\infty = \frac{1}{1 - 1/3} = \frac{1}{2/3} = \frac{3}{2} = \mathbf{1.5}$

⚛️ Physics Use:

Multiple reflections in optics (finding total intensity of light bouncing between two parallel mirrors), Electrostatics problems with image charges — all use the infinite GP sum formula.

§5. Componendo and Dividendo

RULE

If $\frac{a}{b} = \frac{p}{q}$, then by Componendo & Dividendo:

$$\frac{a + b}{a - b} = \frac{p + q}{p - q}$$

This rule allows you to manipulate ratio equations without full cross-multiplication.

Solved Example: If $\frac{x + 1}{x - 1} = \frac{3}{2}$, find $x$.

Using Componendo & Dividendo (comparing $\frac{a}{b} = \frac{3}{2}$ where $a = x + 1, b = x - 1$):
$\frac{(x + 1) + (x - 1)}{(x + 1) - (x - 1)} = \frac{3 + 2}{3 - 2} \implies \frac{2x}{2} = \frac{5}{1} \implies \mathbf{x = 5}$

Practice Questions

DRILL 1 — INDICES & SURDS

1. Simplify: $\frac{x^5 \cdot x^{-2}}{x^3}$
2. Evaluate: $(27)^{2/3}$
3. Simplify: $\frac{(a^2 b^3)^2}{a^3 b^4}$
4. If $2^x = 16$, find $x$.
5. Simplify: $\left(\frac{x^{-1} y^2}{x^2 y^{-1}}\right)^3$

DRILL 2 — QUADRATIC EQUATIONS

6. Solve: $x^2 - 5x + 6 = 0$
7. Solve: $2x^2 + x - 6 = 0$
8. Find the discriminant of $3x^2 - 4x + 2 = 0$ and state the nature of roots.
9. If the roots of $x^2 + px + q = 0$ are 3 and 4, find $p$ and $q$.
10. Form a quadratic equation whose roots are $-2$ and $5$.

DRILL 3 — BINOMIAL APPROXIMATION

11. Approximate $(1 + 0.01)^{10}$ using binomial theorem.
12. Approximate $\frac{1}{\sqrt{1 - 0.04}}$ to first order in $x$.
13. If $g' = g\left(1 - \frac{2h}{R}\right)$, derive this using binomial approximation from $g' = g\frac{R^2}{(R+h)^2}$.
14. Approximate $(0.99)^8$.
15. Show that for small $x$: $(1 - x)^{-1} \approx 1 + x$

DRILL 4 — PROGRESSIONS

16. Find the 15th term of the AP: $3, 7, 11, 15, \dots$
17. The sum of first 10 terms of an AP is 155. If the first term is 2, find the common difference.
18. Find the 6th term of the GP: $2, 6, 18, 54, \dots$
19. Find the sum to infinity: $8 + 4 + 2 + 1 + \dots$
20. Insert 3 arithmetic means between 5 and 25.

DRILL 5 — MIXED & ADVANCED

21. A ball is thrown such that $s = 20t - 5t^2$. Find the times when it returns to the ground ($s=0$).
22. The error in a length measurement is 2%. Using the binomial approximation, find the error in volume (cubical).
23. Sum the infinite series: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$
24. If $\frac{v+u}{v-u} = \frac{3}{1}$, find $v$ in terms of $u$ using Componendo & Dividendo.
25. The product of two consecutive integers is 182. Set up and solve the quadratic equation to find them.
26. A capacitor discharges such that charge $q = q_0 r^t$ where $0 < r < 1$. Find total charge that flows (assuming $t$ starts from 0 to $\infty$ in annual steps).
27. Solve: $\frac{2x-1}{x+3} = \frac{x-2}{x-1}$
28. What is the sum of first $n$ natural numbers? Derive using AP sum formula.
29. If roots of $3x^2 + kx + 12 = 0$ are equal, find $k$.
30. Simplify without a calculator: $\frac{(2^3)^2 \cdot 4^{-1}}{8^{2/3}}$

Answer Key

Q	Answer & Method
1	$x^0 = \mathbf{1}$. Powers: $5 + (-2) - 3 = 0$.
2	$(27)^{2/3} = (\sqrt[3]{27})^2 = 3^2 = \mathbf{9}$.
3	$\frac{a^4 b^6}{a^3 b^4} = \mathbf{ab^2}$.
4	$2^x = 2^4 \implies \mathbf{x = 4}$.
5	$\left(\frac{y^3}{x^3}\right)^3 = \mathbf{\frac{y^9}{x^9}}$.
6	Factor: $(x - 2)(x - 3) = 0 \implies \mathbf{x = 2, 3}$.
7	$\mathbf{x = 3/2} \text{ or } \mathbf{x = -2}$. [Use quadratic formula or factor as $(2x - 3)(x + 2) = 0$]
8	$\Delta = 16 - 24 = \mathbf{-8} < 0$. No real roots.
9	$p = -(3 + 4) = \mathbf{-7}, \quad q = 3 \times 4 = \mathbf{12}$.
10	$(x + 2)(x - 5) = 0 \implies \mathbf{x^2 - 3x - 10 = 0}$.
11	$\approx 1 + 10(0.01) = \mathbf{1.10}$. (Exact: 1.10462)
12	$(1 - 0.04)^{-1/2} \approx 1 + \frac{1}{2}(0.04) = \mathbf{1.02}$.
13	$\frac{R^2}{(R+h)^2} = \left(1 + \frac{h}{R}\right)^{-2} \approx 1 - \frac{2h}{R}$. So $g' \approx g\left(1 - \frac{2h}{R}\right)$. $\checkmark$
14	$(1 - 0.01)^8 \approx 1 + 8(-0.01) = \mathbf{0.92}$.
15	Show that for small $x$: $(1 - x)^{-1} \approx 1 + x$ $\checkmark$
16	$a_{15} = 3 + 14(4) = \mathbf{59}$.
17	$155 = \frac{10}{2}(4 + 9d) \implies d = \mathbf{3}$.
18	$a_6 = 2 \cdot 3^5 = 2 \times 243 = \mathbf{486}$.
19	$a = 8, \ r = 1/2. \ S_\infty = \frac{8}{1-0.5} = \mathbf{16}$.
20	Means: 10, 15, 20. [AP: 5, 10, 15, 20, 25 with $d=5$]
21	$5t^2 - 20t = 0 \Rightarrow t(5t-20)=0 \Rightarrow t = \mathbf{0s, 4s}$.
22	Error in volume $\approx 3 \times 2\% = \mathbf{6\%}$ (binomial: $(1+0.02)^3 \approx 1+0.06$).
23	$a=1/2, r=1/2$. $S_\infty = \frac{0.5}{0.5} = \mathbf{1}$.
24	Comp & Divid: $\frac{2v}{2u}=\frac{4}{2} \Rightarrow v = \mathbf{2u}$.
25	$n(n+1)=182 \Rightarrow n^2+n-182=0 \Rightarrow n=\mathbf{13}$ (integers: 13, 14).
26	Infinite GP with $a=q_0, r=r$. $S_\infty = \mathbf{\frac{q_0}{1-r}}$.
27	$(2x-1)(x-1)=(x-2)(x+3) \Rightarrow \mathbf{x = 7 \text{ or } -1}$.
28	$S_n = \frac{n}{2}[2(1)+(n-1)(1)] = \mathbf{\frac{n(n+1)}{2}}$.
29	$\Delta = 0 \Rightarrow k^2 - 4(3)(12)=0 \Rightarrow k = \pm\mathbf{12}$.
30	$\frac{64 \cdot \frac{1}{4}}{4} = \frac{16}{4} = \mathbf{4}$.