IB Maths AA HL Topic 1 — Number & Algebra Paper 1 & 2 ~6 min read

Standard Form

Standard form is just a tidy way of writing very big or very small numbers. The IB requires you to write your final answer in this form when asked, so the trick isn’t really the maths — it’s spotting when a number isn’t in the correct shape, and fixing it. Your GDC will handle most of the heavy lifting; you just need to read its output properly.

📘 What you need to know

Standard form is a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
For large numbers the power n is positive; for small numbers it’s negative.
To multiply or divide: handle the a‘s separately, handle the powers of 10 separately (add or subtract the indices).
To add or subtract: match the powers of 10 first, then combine the a‘s.
Always check the final answer is in the form 1 ≤ a < 10. If a is 12 or 0.4, your answer isn’t in standard form yet.
This is not in the formula booklet — it’s notation, so just learn it.

What does standard form actually mean?

Some numbers are too big or too small to write out comfortably. The mass of the Earth is roughly 5,972,000,000,000,000,000,000,000 kg. The diameter of a hydrogen atom is about 0.000000000106 m. Counting zeros for either is a recipe for losing marks. Standard form pulls that whole mess into something compact.

Standard form a × 10ⁿ, where 1 ≤ a < 10 and n ∈ ℤ ✗ not in the formula booklet — memorise the form

🌍 Real-world feel

Earth’s mass ≈ 5.97 × 10²⁴ kg. Hydrogen atom ≈ 1.06 × 10⁻¹⁰ m. Both numbers fit on one line, the powers tell you the scale at a glance, and you don’t have to count zeros twice.

Writing numbers in standard form

The whole game is moving the decimal point until exactly one non-zero digit sits in front of it. Then count how many places you moved.

Where the decimal point lands

Quick mental check: if the original number is bigger than 10, the power is positive. If it’s smaller than 1, the power is negative. If it’s between 1 and 10, the power is 0.

Multiplying and dividing

This is the part students like — the powers of 10 obey the index laws, and the leading numbers just multiply or divide as normal.

Multiply

(a1 × 10^m) × (a2 × 10ⁿ)

= (a1 × a2) × 10^{m + n}

Divide

(a1 × 10^m) ÷ (a2 × 10ⁿ)

= (a1 ÷ a2) × 10^{m − n}

After multiplying or dividing the leading parts, check that the result still satisfies 1 ≤ a < 10. If it doesn’t, shift the decimal and adjust the power. That last shuffle is where almost all the marks get lost.

🧭 Recipe — multiply or divide in standard form

Group: separate the leading numbers from the powers of 10.
Operate: multiply (or divide) the leading numbers; add (or subtract) the indices.
Standardise: if the new a isn’t between 1 and 10, write it in standard form, then combine the powers of 10 again.

Adding and subtracting

This is where students slow down. You can’t just add the leading numbers — the powers of 10 must match first.

🧭 Recipe — add or subtract in standard form

Pick the bigger power of 10 and rewrite the smaller-power term so its 10ⁿ matches.
Now add or subtract the leading numbers — the powers of 10 are just a common factor.
Standardise the final answer so 1 ≤ a < 10.

🤔 Why match powers first?

Because 10⁵ and 10⁴ are different units of size. Adding “3 millions” and “5 thousands” only works if you first convert one of them — same idea here. Pull a common power of 10 out as a factor, then add what’s left.

Negative powers — same logic: 10⁻⁶ is bigger than 10⁻⁹. The “highest” (least negative) power wins. Match the smaller-power term up to it before combining.

Worked examples

WE 1

Convert to and from standard form

(i) Write 92 400 in standard form. (ii) Write 6.7 × 10⁻⁵ as an ordinary number.

(i) 92 400 → standard form move decimal 4 places left: 9.2400 9.24 × 10⁴ (ii) 6.7 × 10⁻⁵ → ordinary negative power → small number; move decimal 5 places left 0.000067

WE 2

Multiplication that needs standardising

Calculate (4 × 10⁷) × (3 × 10⁵), giving the answer in standard form.

Step 1: Group leading numbers and powers = (4 × 3) × (10⁷ × 10⁵) = 12 × 10¹² Step 2: Standardise — 12 is not between 1 and 10 12 = 1.2 × 10¹ so 12 × 10¹² = 1.2 × 10¹ × 10¹² = 1.2 × 10¹³ forgetting to standardise the leading number is the #1 mark-losing slip

WE 3

Division giving a negative power

Calculate (3 × 10²) ÷ (8 × 10⁶), giving the answer in standard form.

Step 1: Group = (3 ÷ 8) × (10² ÷ 10⁶) = 0.375 × 10^{2 − 6} = 0.375 × 10⁻⁴ Step 2: Standardise — 0.375 is less than 1 0.375 = 3.75 × 10⁻¹ so 0.375 × 10⁻⁴ = 3.75 × 10⁻¹ × 10⁻⁴ = 3.75 × 10⁻⁵

WE 4

Addition with different powers

Calculate (6 × 10⁸) + (4 × 10⁷), giving the answer in standard form.

Step 1: Match the powers — pick 10⁸ 4 × 10⁷ = 0.4 × 10⁸ Step 2: Add the leading numbers (6 × 10⁸) + (0.4 × 10⁸) = 6.4 × 10⁸ = 6.4 × 10⁸ already in standard form — no extra adjustment needed

WE 5

Subtraction with negative powers

Calculate (5 × 10⁻⁶) − (2 × 10⁻⁷), giving the answer in standard form.

Step 1: Spot the bigger power — 10⁻⁶ is bigger than 10⁻⁷ 2 × 10⁻⁷ = 0.2 × 10⁻⁶ Step 2: Subtract (5 × 10⁻⁶) − (0.2 × 10⁻⁶) = 4.8 × 10⁻⁶ = 4.8 × 10⁻⁶ −6 is “bigger” than −7 — number-line thinking, not absolute value

WE 6

Application — light travel time

Light travels at approximately 3 × 10⁸ m/s. The Sun is approximately 1.5 × 10¹¹ m from Earth. How long, in seconds, does sunlight take to reach Earth? Give your answer in standard form.

Step 1: time = distance ÷ speed t = (1.5 × 10¹¹) ÷ (3 × 10⁸) Step 2: Group and divide = (1.5 ÷ 3) × 10^{11 − 8} = 0.5 × 10³ Step 3: Standardise — 0.5 is not between 1 and 10 0.5 × 10³ = 5 × 10² t = 5 × 10² seconds (≈ 8.3 min)

💡 Top tips

Always re-check that 1 ≤ a < 10 at the end. Most “wrong” answers are arithmetically correct but sit one decimal jump out of standard form.
Use scientific mode (SCI) on your GDC. Once you’re in SCI mode, the calculator outputs every answer in standard form automatically — saves you the standardising step.
The GDC display “6.5E5” means 6.5 × 10⁵. Different brands show it differently (some use ×10 with a small power); just write it properly on paper.
For add/subtract, write both terms with the same power before doing anything else. Don’t try to do it in your head.
If the answer comes out clean (like 1.2 × 10¹³), don’t second-guess it — IB problems often are set up to give round numbers.
If the question says “give your answer in the form a × 10ⁿ“, that’s a direct instruction — losing a mark for skipping it is genuinely painful.

⚠ Common mistakes

Leaving the leading number outside the 1 ≤ a < 10 range. Writing 12 × 10¹² as a final answer is not standard form.
Adding before matching powers. (6 × 10⁸) + (4 × 10⁷) ≠ 10 × 10¹⁵. The powers must match first.
Confusion with negative powers. 10⁻⁶ is larger than 10⁻⁹. Bigger negative number = smaller value.
Wrong direction for the decimal point. Big numbers → decimal moves left, power positive. Small numbers → decimal moves right, power negative.
Forgetting to adjust the power after standardising the leading number. If 0.375 becomes 3.75 × 10⁻¹, that −1 must combine with the existing power of 10.
Misreading the GDC. “5.6E−4” is 5.6 × 10⁻⁴, not 5.6 − 4. The “E” stands for “× 10 to the…”.

Standard form looks small, but it’ll show up in nearly every paper — Paper 2 problems with very large or very small physical quantities, and any time IB tells you to give your answer to 3 significant figures. Get fluent on it now and you’ll never lose marks here.

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