1The idea in one paragraph

Some numbers have too many zeros to write sensibly: the distance from the Earth to the Sun is about 150 000 000 000 metres, and an atom is about 0.000 000 000 1 metres across. Scientific notation writes every positive number as a number between 1 and 10 multiplied by a power of ten: 1.5 × 10¹¹ and 1 × 10⁻¹⁰. The first part carries the digits and the power carries the size. Because the power does the counting, you can see at a glance which of two numbers is bigger, and you can multiply and divide by working with the two parts separately.

2What the form means

A number is in the form a × 10ᵏ when two conditions hold.

a × 10ᵏ, where 1 ≤ a < 10 and k is an integer (k ∈ ℤ)

a has exactly one non-zero digit before the decimal point. So 3.844, 1 and 9.99 are allowed; 38.44, 0.3844 and 10 are not. k is a whole number, positive, negative or zero. Figure 1 labels the two parts.

The sign of k tells you the size at once:

Why insist on 1 ≤ a < 10? Because it makes the form unique. The number 384 400 could be written as 38.44 × 10⁴ or 0.3844 × 10⁶, and both are true, but only 3.844 × 10⁵ is in the form. One number, one way of writing it, and that is what lets you compare numbers by looking at k.

A negative number keeps its minus sign in front: −0.000 62 = −6.2 × 10⁻⁴. The condition applies to the size of a.

Figure 2 shows why this form is so useful. Each tick on the ruler is ten times the one before, so lengths from an atom to a light year fit on one line. The power k tells you which step a length sits in; a tells you where inside that step.

The figure uses real, rounded values: an atom is about 1 × 10⁻¹⁰ m across, the mean distance from the Earth to the Sun is about 1.5 × 10¹¹ m, and a light year is about 9.5 × 10¹⁵ m. Some numbers are famous enough to have names of their own: a googol is 10¹⁰⁰. Most never get one, and a × 10ᵏ is how they are written instead.

3Converting into the form, and back

Into the form. Move the decimal point until it sits just after the first non-zero digit, and count how many places it moved. That count is the size of k. The direction gives the sign: a number bigger than 10 needs a positive k, a number less than 1 needs a negative k. Figure 3 shows both cases hop by hop.

Check each answer by asking "is it big or small?" before you write the sign of k. That one question catches most slips.

Back to an ordinary number. Run the process backwards: a positive k moves the point k places right, a negative k moves it k places left, filling empty places with zeros.

Fixing a number that is almost in the form. Questions often produce something like 42 × 10⁵ or 0.6 × 10⁻³ halfway through a calculation. Rewrite a, and let the power absorb the change. Making a ten times smaller means making the power ten times bigger, so k goes up by 1; making a ten times bigger sends k down by 1.

Writing the middle step, 4.2 × 10¹ × 10⁵, is the safe way to do this. It shows the examiner your method and it stops you moving k the wrong way.

4Comparing and ordering

For positive numbers in the form, compare k first. A larger k always means a larger number, whatever the a's are, because a can never reach 10. Only if the k's are equal do you compare the a's.

Put these in order, smallest first: 9.9 × 10⁻⁴, 1.2 × 10⁻³, 8.5 × 10⁻⁴, 3 × 10⁻⁵.

The powers are −4, −3, −4 and −5. The smallest power is −5, so 3 × 10⁻⁵ comes first. The two with k = −4 are ordered by a: 8.5 before 9.9. The largest power, −3, comes last.

3 × 10⁻⁵ < 8.5 × 10⁻⁴ < 9.9 × 10⁻⁴ < 1.2 × 10⁻³

Notice that 9.9 × 10⁻⁴ is smaller than 1.2 × 10⁻³ even though 9.9 is bigger than 1.2. With negative powers, remember that −3 is bigger than −4. Write them out as decimals once if you doubt it: 0.000 99 against 0.0012.

5Multiplying, dividing, powers and roots

Multiplication does not care about order, so you can collect the a's together and the powers of ten together. Figure 4 sets out the route on its left side: work with the a's, then with the powers, then tidy up.

The power-of-ten step uses the laws you will meet formally in 1.5: multiplying adds the powers, dividing subtracts them.

The tidy-up step is where the final mark sits. An answer of 24 × 10¹³ is correct in value but not in the form asked for, and it loses the accuracy mark.

Powers. Raise both parts to the power, remembering that (10ᵏ)ⁿ = 10ᵏⁿ.

Square roots. The square root of 10ᵏ is a whole power of ten only when k is even. So first rewrite the number with an even power, then take the root of each part.

On Paper 1 the numbers are chosen so this comes out cleanly. If it does not, check whether you moved k the right way.

6Adding and subtracting

Here the powers cannot simply be combined. 5 × 10⁶ + 3 × 10⁵ is not 8 × 10¹¹ or 8 × 10⁶; the two numbers are counting in different-sized units. Follow the right side of Figure 4: rewrite one number so both have the same power of ten, then add or subtract the a's, then tidy up.

Either number can be the one you rewrite. Rewriting the smaller one to match the larger one usually saves the tidy-up step. If the two powers are far apart, the smaller number may barely change the answer: 4 × 10⁹ + 3 × 10² = 4.000 000 3 × 10⁹, which is 4.00 × 10⁹ to three significant figures.

7With a calculator: Paper 2

A GDC works in this form happily, but it displays it in its own shorthand. 4.73E18 on the screen means 4.73 × 10¹⁸. That shorthand is calculator notation, and the IB does not accept it as an answer. Figure 5 shows the translation you must make every time.

Three habits make Paper 2 calculations safe.

Enter the number with the calculator's own ×10ⁿ key (labelled EE or ×10ˣ, depending on the model) rather than typing × 10 ^. The key keeps the number together as one value, so a division by 6.02 × 10²³ divides by the whole number, not by 6.02 and then multiplies by 10²³.

Keep full accuracy until the end. Store intermediate values in the memory; round only the final answer.

Round a, not the whole number. "Three significant figures" in this form means three digits in a: 6.923 076… × 10⁹ becomes 6.92 × 10⁹. The IB's default, when a question does not say otherwise, is an exact answer or three significant figures.

Two worked examples with real, rounded constants from physics and chemistry.

How long does light take to reach us from the Sun? The mean Earth–Sun distance is about 1.496 × 10¹¹ m and the speed of light is about 2.998 × 10⁸ m s⁻¹.

What is the mass of one carbon-12 atom? A mole of carbon-12 has mass 12 g and contains about 6.022 × 10²³ atoms (the Avogadro constant, which you will meet in chemistry).

Physics also talks about the order of magnitude of a quantity: the power of ten it is nearest to. A light year, 9.5 × 10¹⁵ m, is of order 10¹⁶ m. That is a useful estimate and a quick check on any calculator answer: if the Sun's light seemed to take 10⁻³ seconds to arrive, something was typed wrongly.