Physics is the study of nature, and the first step to understand nature is to measure it. Think of a tailor shop — the tailor never guesses, he measures your cloth with a metre tape. That's exactly what measurement is: comparing an unknown quantity with a known standard.

That known standard is called a unit. So, every measurement has two parts — a number and a unit. We write it as: Measurement = n × u.

In daily life we do this all the time. We buy cloth in metres, vegetables in kilograms from the sabzi mandi, and we note cooking time in seconds. All these are called physical quantities because they can be measured.

Without units, communication breaks down. If you tell someone to bring '2 sugar', it's confusing — 2 spoons? 2 packets? But if you say '2 kg sugar', it's crystal clear. Units make our language and science precise.

Just like VIBGYOR has 7 base colours and mixing them gives you every other colour, physics has 7 fundamental base quantities. They are independent — you cannot make one from another. All other quantities are derived from them.

Let's meet them with daily life examples:

1. Length – metre (m): What the tailor measures for your kurta.
2. Mass – kilogram (kg): What the sabziwala uses to weigh aloo on a tarazu.
3. Time – second (s): One heartbeat is roughly one second. Tick-tock.
4. Temperature – kelvin (K): What a thermometer shows when you have fever.
5. Electric Current – ampere (A): Flow of bijli, like water flowing in a pipe to glow a bulb.
6. Luminous Intensity – candela (cd): Brightness of a torch at night.
7. Amount of Substance – mole (mol): A way of counting very small things. Like a dozen eggs is 12, one mole is 6.022 × 10²³ particles — just a bigger counting unit, like counting laddoos in bulk.

Any other quantity like speed (m/s), area (m²) or volume (m³) is derived by mixing these 7 base units.

Over the years, people used different systems to measure.

CGS System (cm, g, s): For small lab work — like measuring your pinky nail or a spoon of sugar.
FPS System (foot, pound, second): The old British system. Think of your British uncle measuring a foot-long bread loaf. Still used in US/UK for some things.
MKS System (metre, kg, second): The 'Bada Bazaar' system — practical for big daily work like measuring cloth in metres and sabzi in kg.
SI System (International): The modern, universal system accepted worldwide. It has 7 base units plus 2 supplementary units — radian (rad) for plane angle, like a pizza slice, and steradian (sr) for solid angle, like the cone of light from a torch. SI is coherent and logical.

How are base units now defined?
Metre: Distance light travels in vacuum in 1/299792458 seconds. Light is the fastest courier in universe.
Kilogram: No longer a metal cylinder in Paris vault. Now defined by fixing Planck's constant h = 6.626 × 10⁻³⁴ J·s — a universal constant.
Second: Defined by 9,192,631,770 vibrations of a Cesium-133 atom. Imagine a tiny tabla beating super fast inside the atom — we count those beats.

Have you noticed parallax? Hold your thumb in front of you. Close your left eye, your thumb appears on the right side of the background. Close your right eye, it shifts to the left. That apparent shift when you move your eye is called parallax.

We see it daily — when you're in an auto, nearby trees seem to move fast while far buildings barely move.

This trick is used to measure huge distances. Formula is simple: Parallax angle = θ (theta), baseline = b (distance between two observation points, like distance between your two eyes ~6 cm). Then distance to object D = b / θ, where θ must be in radians. If θ is very small, D becomes very large.

For far stars, we use Earth's orbit as baseline. We observe a star from location A, then six months later from location B (opposite side of Earth). The shift gives us distance to Moon and stars.

Range of lengths in universe is mind-blowing: from 10⁻¹⁵ m (size of a proton/nucleus) to 10²⁶ m (size of observable universe). For tiny lengths, we can't use metre scale. We use Screw Gauge (up to 10⁻⁵ m) and Vernier Calipers / Microscope (up to 10⁻⁶ m).

Mass has a huge range. From an electron which is 9.1 × 10⁻³¹ kg — imagine a single til ka tukda — to the entire universe which is about 10⁵⁵ kg, like the whole sabzi mandi of existence.

To measure tiny masses, we define atomic mass. 1 amu = 1/12th mass of a Carbon-12 atom. Think of it like 1 idli out of a plate of 12 idlis being the standard.

The standard kilogram was once a metal cylinder locked in a vault in Paris. Now it's defined by nature using Planck's constant h = 6.626 × 10⁻³⁴ J·s, so it's fixed forever.

Two types of mass:
Inertial Mass (m = F/a): How much an object resists being pushed. Like pushing a loaded auto rickshaw vs pushing a lazy uncle on a sofa — more inertial mass is harder to move.
Gravitational Mass (W = m·g): What a bathroom weighing machine shows — how strongly Earth pulls you.

How do we measure? Very small (gold, jeera) → Goldsmith's tarazu. Large (truck loaded with grains) → Weighing bridge / kata.

Now Time. Its range is also huge: 10⁻²³ s is time light takes to cross a nucleus, 1 second is roughly one dil ki dhadkan, and 10¹⁸ s is age of universe — 13.8 billion years.

Clocks evolved: Pendulum clock (jhula) — one swing ~2 sec. Quartz clock — quartz crystal vibrates 32,768 times per second. Atomic clock (Cesium) — uses 9,192,631,770 vibrations of Cesium atom — most accurate clock we have.

Imagine shooting arrows at a target.

Accuracy means how close you are to the TRUE value — the bullseye. Precision means how close your shots are to each other — like making same-size rotis every time.

So we have 4 cases:
1. High Accuracy, High Precision: Perfect — all arrows in bullseye together.
2. Low Accuracy, High Precision: Roti is perfectly round same size but made at wrong place — shots clustered together but away from bullseye.
3. High Accuracy, Low Precision: Average is bullseye but shots scattered.
4. Low Accuracy, Low Precision: Completely confused — scattered and away.

Errors are why we miss:
Systematic Error: Faulty instrument — like a weighing scale that always shows 1kg extra, or zero error in vernier calipers.
Random Error: Unpredictable — like wind shaking your hand while measuring.
Gross Error: Human mistake — reading wrong or careless.

We quantify errors:
Absolute Error = |True value – Measured value|
Mean Absolute Error = average of all absolute errors.
Relative Error = Mean Absolute Error / True value.
Percentage Error = Relative Error × 100%.

When we add, subtract, multiply, divide, or raise to power, errors also propagate. Arjuna needed both accuracy and precision to hit the fish's eye — same in experiments.

Significant figures are the meaningful digits in a measurement that tell us about its precision. It's like counting REAL voting members in a joint family — only those who actually count, not the crowd standing outside.

Rules to remember:

1. All non-zero digits are significant. Like Dadaji and Mummy — always counted. 325 has 3 sf.
2. Zeros between non-zero digits are significant. Like a child standing between two parents — safe and counted. 2005 has 4 sf.
3. Trailing zeros WITH decimal point are significant. Like youngest child with Aadhar ID — confirmed identity. 2.50 has 3 sf, 2.5 has only 2 sf.
4. Leading zeros are NOT significant. Like guests waiting outside gate — not counted yet. 0.0025 has only 2 sf.
5. Trailing zeros WITHOUT decimal are AMBIGUOUS. Like strangers at a wedding — are they family or not? 2500 could be 2, 3 or 4 sf. Better write in scientific notation: 2.5 × 10³ = 2 sf, 2.50 × 10³ = 3 sf.

Operations:
Addition/Subtraction: Result should have same number of decimal places as term with least decimal places. Ex: 2.5 + 1.234 = 3.7 (not 3.734).
Multiplication/Division: Result should have same number of significant figures as term with least significant figures. Ex: 2.5 × 1.234 = 3.1 (since 2.5 has 2 sf).

Rounding: If digit to be dropped >5, increase previous; if <5, keep; if =5, check even/odd rule.

Dimensions are like the DNA or Aadhar card of a physical quantity. They don't tell you how big it is, but they tell you what it is REALLY made of.

For example, speed looks like distance/time. So its DNA is Length¹ × Time⁻¹. We write it as [M⁰L¹T⁻¹].

Think of the 7 base quantities as 7 grandparents or ancestors. Every physical quantity in the universe is built from them, just like your surname shows your lineage.

The 7 bases are:
M = Mass (weighing machine)
L = Length (ruler)
T = Time (clock)
K = Temperature
A = Electric Current
cd = Luminous intensity
mol = Amount

We write dimensions in square brackets as: [M^a L^b T^c ...]

Some examples:
Speed = Distance/Time = L/T = [M⁰L¹T⁻¹] — like an auto rickshaw moving.
Acceleration = Speed/Time = [M⁰L¹T⁻²] — scooty picking up speed.
Force = Mass × Acceleration = [M¹L¹T⁻²] — pushing an almirah.
Work / Energy = Force × Distance = [M¹L²T⁻²] — carrying a heavy bag.
Power = Work/Time = [M¹L²T⁻³] — your fast charger.
Pressure = Force/Area = [M¹L⁻¹T⁻²] — sharpness of a needle.

Quantities that have no dimensions: angle, strain, refractive index — they are pure numbers.

Think of dimensions as an ID card [M^a L^b T^c] or a recipe's ingredients list. Dimensional analysis has 3 superpowers:

1. Checking Correctness: Principle of Homogeneity — you can only add or equate apples with apples. You cannot add length to time. So in any correct equation, dimensions of LHS must equal dimensions of RHS.

Example: Check s = ut + ½at².
Dimensions: s = [L], ut = [L·T⁻¹][T] = [L], at² = [L·T⁻²][T²] = [L].
So [L] = [L] + [L] → [L] = [L] — Correct! If it was [L] = [L] + [T], it would be Apple + Orange = WRONG.

2. Deriving Relation: For a pendulum jhula, we guess Time period T depends on length l, gravity g, and mass m. Assume T = k · l^a · g^b · m^c. Compare dimensions of M, L, T on both sides. You get c=0 (mass cancels), a=½, b= -½. So T = k·√(l/g). We learn T doesn't depend on mass!

3. Converting Units: Like converting Rupees to Dollars. Formula: n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c.

Example: Convert 1 Newton to dyne. 1 N = 1 kg·m/s². In CGS, it's g·cm/s² = dyne.
So n₂ = 1 × [1000g/1g]¹ [100cm/1cm]¹ = 10⁵. So 1 N = 10⁵ dyne. It's just recipe scaling!

Tip to remember: Apples = Checking, Jhula = Deriving, Currency = Converting.

Dimensional analysis is powerful, but it has limits. Think of it as an X-ray machine.

1. X-ray can see bones but not colour: An X-ray machine can see bones [M, L, T] but cannot see if your t-shirt is red or blue, or its design. Similarly, dimensions cannot see dimensionless numbers like ½, 2, π, or functions like sin, cos, exp, log.

2. Cannot find the value of dimensionless constant k (Missing Tadka): Formula s = ut + ½at². Dimensional analysis will only tell you s = ut + K·at², but it cannot tell you that K = ½. Like a recipe tells you 2 cups atta (dimensions) but cannot tell you chutki bhar haldi — the exact pinch.

3. Cannot handle sin, cos, exp, log: It's colour blind. If s = A sin(wt) or y = e^x, argument of sin/exp must be dimensionless, but method fails to check formulas with such functions. X-ray sees shirt size but not its colour.

4. Cannot distinguish between quantities with same dimensions (Twins with same height): Work = Force × distance (scalar) and Torque = r × F (vector) both have same dimensions [M¹L²T⁻²] but are totally different — one is lifting a box, other is opening a door with a wrench. Dimensions can't tell twins apart.

Despite limitations, we use it for Order of Magnitude estimation — quick rough calculation of size. Like estimating number of drops in a water tank or number of pages in this chapter. It's about getting the power of 10 correct.