Physics · Class 9-10 · Chapter 1 · Physical quantities and measurement
Prefixes, symbols and scientific notation.
The numbers in physics are either absurdly large or absurdly small. Count the zeros by hand and you will miscount. So there are two devices, and both rest on one idea: stop writing zeros and start writing the power.
1. The problem
Writing the mass of the earth means writing 5972000000000000000000000 kg. Count the zeros, and notice that miscounting by one changes the answer tenfold. At the other end, the radius of an atom is about 0.0000000001 m. Both are painful to write, painful to read, and very nearly impossible to multiply.
The fix is scientific notation: write any number as a × 10ⁿ, where the front number a lies between 1 and 10 and n is a whole number. The mass of the earth becomes 5.972 × 10²⁴ kg and the radius of an atom becomes 1 × 10⁻¹⁰ m. The zeros have moved into the exponent.
The rule is one line: move the point left and the exponent goes up, move it right and the exponent goes down, one step for each place. Rather than memorising that, move the point yourself below. Then the sign stops flipping on you.
move it yourselfmove the decimal point
Instead of counting zeros
2. Prefixes
The second device is more familiar. Take the power and put it in front of the unit as a word, and that word is a prefix. Instead of 1000 metres you write 1 kilometre, because kilo just means 10³. You use these every day: GB, MHz, mA, mg, cm are all units with a prefix on the front.
Going up: deca da 10¹, hecto h 10², kilo k 10³, mega M 10⁶, giga G 10⁹, tera T 10¹², peta P 10¹⁵, exa E 10¹⁸. Going down: deci d 10⁻¹, centi c 10⁻², milli m 10⁻³, micro μ 10⁻⁶, nano n 10⁻⁹, pico p 10⁻¹², femto f 10⁻¹⁵, atto a 10⁻¹⁸.
Twelve prefixes below. Sort them into the two directions, and while you do it the pattern in the capital and small letters will start to show.
sort themchip, then bin
Up or down?
Pick up a prefix, then drop it in a bin.
Right. Next one.
Not there. Look at the sign of the exponent: positive is up, negative is down.
Now look at the symbols. From mega upward they are capitals (M, G, T), and everything on the way down is lower case. Kilo (k) is lower case, and that is the one exception you have to remember.
There is a trap right here, and somebody falls into it every time. Capital M is mega, 10⁶. Lower case m is milli, 10⁻³. A factor of 10⁹ between them. 1 MW is a million watts; 1 mW is a thousandth of one watt. And one more rule: the prefix goes in front of the unit, not in front of the number. 5 km is right, 5k m is not.
Going from one prefix to another needs nothing memorised, only the two exponents subtracted. Change the boxes below and watch the working fall into three lines.
calculatorchange the number or a prefix
The prefix converter
check yourselfpick one answer
The diameter of a wire
A wire is 0.2 mm across. How many micrometres (μm) is that?
Right. Subtract the exponents: mm is 10⁻³ and μm is 10⁻⁶, so −3 − (−6) = 3. That gives 0.2 × 10³ = 200 μm.
This used 2 for the exponent instead of 3. From mm to μm is a factor of a thousand, not a hundred.
One power too many. 0.2 × 10³ is 200, not 2000. The 0.2 in front has to be carried through the count.
This divided where it should have multiplied. A micrometre is a smaller unit than a millimetre, so the same length gets a bigger number, not a smaller one.
3. One step is ten times
Reading that “physics runs from 10⁻¹⁵ m to 10²⁶ m” in a paragraph tells you nothing. Dragging a handle across it tells you something. Keep in mind that every notch on the handle is a factor of ten, and two notches is a hundred.
handledrag the handle
The powers of ten handle
The drawing is a model, and the sizes are approximate.
Forty one notches of a handle took you from inside a nucleus to the edge of the visible universe. That span is what people mean by the scope of physics, and the same handful of laws covers both ends of it. The sizes used here are approximations rather than exact textbook figures.
Reading exponents means being able to compare, and comparing is where one particular mistake keeps happening. Put the seven sizes below in order, smallest first.
smallest firsttap them smallest to largest
From small to large
Start with the smallest.
Yes. Which is next?
Not yet. Look at the exponent first, not the number in front.
Were you reading the number in front? You never needed it. Compare the exponents first, and only look at the front number when the exponents match. 7 × 10⁻⁵ is smaller than 2 × 10⁻², even though 7 is the bigger digit.
4. The symbols for quantities
Every quantity has a standard symbol, and it is the same everywhere in the world. Length l, mass m, time t, velocity v, acceleration a, force F, work W, power P, area A, volume V, temperature T, current I, resistance R, potential difference V. Look at the letters in a formula and you know what it is about, whatever language it is written in.
There is one problem: the letters run out. So the same letter ends up standing for two quantities. V is volume and also potential difference. T is time period and also temperature. You work it out from context. There is also a small convention that helps: symbols for quantities are set in italic, symbols for units in roman. So m is mass and m is a metre.
When the Roman letters run out, Greek ones take over: ρ density, λ wavelength, θ angle, ω angular velocity, μ coefficient of friction, Δ change, Σ sum, π 3.1416, and α β γ for the radiations. Tap the symbols below for what each one means and where it can be confused.
symbolstap a symbol
What is this symbol?
v Velocity, in m/s. Note the small v; a capital V is something else entirely.
V Volume (m³), or potential difference (V). Context decides: if the formula is about electricity, it is voltage.
a Acceleration, in m/s². As a prefix, a lower case a means atto, 10⁻¹⁸.
F Force, in newtons (N). The formula F = ma uses three of the symbols on this row.
W Work, in joules (J). As a unit, W means the watt. Quantity symbols are italic and unit symbols roman, and this pair is why that matters.
P Power, in watts (W). Pressure is sometimes P as well, and then the unit is the pascal (Pa).
ρ Rho. Density, in kg/m³. Not an English p, a Greek letter.
λ Lambda. Wavelength, in metres. It comes back all through light and sound.
θ Theta. An angle, measured in degrees or radians.
ω Omega. Angular velocity, in rad/s. The capital Ω, though, is the ohm, a unit of resistance.
Δ Delta. It means “change in”. Δv is a change of velocity, not a quantity of its own. It never appears alone.
μ Mu. The coefficient of friction, which has no unit at all. As a prefix, μ means micro, 10⁻⁶. One letter, two jobs.
All twelve. Notice that nearly every place these get confused is a difference between a capital and a small letter.
Writing unit symbols has rules of its own, and breaking them costs marks. A unit named after a person takes a capital symbol (N, J, W, A, K, Pa, Hz) but a lower case name when written out (newton, joule). No full stop after a symbol, and no plural s. 5 kg is right; 5 kgs and 5 kg. are not.
check yourselfpick one answer
Which one follows the rules
Which of these does not break a rule about writing unit symbols?
Right. Lower case k for kilo, lower case g for gram, no plural s, and no full stop at the end.
A capital K is the kelvin, a unit of temperature. The prefix kilo is always a lower case k.
Unit symbols never take a plural s. Five kilograms is 5 kg, just as one kilogram is 1 kg.
No full stop after a symbol unless it is also the end of the sentence. Here it reads as though the stop were part of the symbol.
5. Some sizes worth carrying around
A few sizes in your head let you sanity check any answer. The radius of the earth is about 6.4 × 10⁶ m, the mass of the sun about 2 × 10³⁰ kg, the mass of an electron about 9.1 × 10⁻³¹ kg, the radius of an atom about 10⁻¹⁰ m, the radius of a nucleus about 10⁻¹⁵ m, and the acceleration due to gravity g = 9.8 m/s².
They are not there to be memorised, they are there for comparison. If a calculation hands you an atom with a radius of 10⁻⁵ m, something is five powers wrong, and you can catch it yourself.
check yourselfpick one answer
In scientific notation
Written in scientific notation, what is 0.000045 m?
Right. Moving the point five places to the right gives 4.5, so the exponent is −5.
The value is right, since 45 × 10⁻⁶ is 4.5 × 10⁻⁵. But scientific notation requires the front number to lie between 1 and 10, and 45 does not.
One power too many. Count it: in 0.000045 there are four zeros before the 4, so the point moves five places, not six.
The sign is the wrong way round. The number is less than one, so the exponent has to be negative. 4.5 × 10⁵ is four hundred and fifty thousand.
Source: Physics, Classes 9-10 (NCTB), Chapter 1 “Physical quantities and measurement”, the sections on prefixes and on scientific symbols and notation.