IB Maths AI HLNumber ToolkitPaper 1 & 2a × 10n form~6 min read
Standard Form
Standard form writes any number — however huge or tiny — as a single digit-string between 1 and 10 multiplied by a power of 10. This note shows how to convert numbers into that form and how to multiply, divide, add and subtract them while keeping the answer in standard form.
📘 What you need to know
Standard form is a × 10n, where 1 ≤ a < 10 and n is an integer.
n is positive for large numbers, negative for small numbers (less than 1).
n counts how many places the decimal point moves to turn the number into a.
Multiply / divide: handle the a-parts and the powers of 10 separately — add the powers when multiplying, subtract them when dividing.
Add / subtract: rewrite both numbers with the same power of 10 first, then combine the a-parts.
If a result’s a-part falls outside 1 ≤ a < 10, re-write it in standard form and combine the powers.
Writing numbers in standard form
A number in standard form is a × 10n: the value a carries the significant digits and must satisfy 1 ≤ a < 10, while the integer n records the size by counting decimal-point jumps.
Standard forma × 10n, 1 ≤ a < 10, n ∈ ℤ
n > 0 for large numbers — n < 0 for small numbers
For a large number, place the decimal point after the first non-zero digit and count how many places it jumped — that count is n, and it is positive. For a small number (less than 1) the decimal moves the other way, so n is negative.
The decimal point jump sets n: jumping left for a large number gives a positive power, jumping right for a small number gives a negative power.
Multiplying and dividing
To multiply or divide numbers in standard form, deal with the two parts separately: combine the a-parts by ordinary arithmetic, and combine the powers of 10 using the index laws.
Index laws for the powers of 10
10m × 10n = 10m+n10m ÷ 10n = 10m−nadd the powers to multiply, subtract them to divide
If the new a-part is not between 1 and 10 — for example 12 or 0.25 — rewrite that part in standard form and add its power onto the running total.
Adding and subtracting
You cannot add the a-parts directly unless the powers of 10 match. So first find the higher power of 10, then rewrite the other number with that same power — its a-part will no longer be between 1 and 10, and that is fine for this step.
Once both numbers share a power of 10, add or subtract the a-parts and write the result back in standard form.
GDC shortcut: in scientific mode your calculator will multiply, divide, add and subtract numbers in standard form directly and return the answer already in a × 10n form.
🧠Recipe — standard form in five steps
Find a: read off the significant digits as a number between 1 and 10.
Find n: count how many places the decimal point moves to reach a.
Fix the sign of n: positive for large numbers, negative for numbers below 1.
For a calculation: handle the a-parts and the powers of 10 separately (add/subtract powers; match powers before adding numbers).
Tidy up: if the a-part is outside 1 ≤ a < 10, rewrite it in standard form and combine the powers.
Worked examples
WE 1
Writing a large number
Light travels about 9 460 000 000 000 km in one year. Write this distance in standard form.
find a — digit-string between 1 and 10a = 9.46count the decimal jumps to reach a9 460 000 000 000 → 9.46 — jumps 12 places left9.46 × 1012 kmlarge number, so the power is positive.
WE 2
Writing a small number
A red blood cell has a diameter of about 0.0000078 m. Write this length in standard form.
find a — digit-string between 1 and 10a = 7.8count the decimal jumps to reach a0.0000078 → 7.8 — jumps 6 places right7.8 × 10−6 mnumber below 1, so the power is negative.
WE 3
Multiplying in standard form
Evaluate (6 × 104) × (7 × 109), giving your answer in standard form.
multiply the a-parts, add the powers6 × 7 = 42104 × 109 = 101342 is not between 1 and 10 — rewrite it42 = 4.2 × 101 ⇒ 4.2 × 101 × 10134.2 × 1014when the a-part overshoots 10, bump the power up by 1.
WE 4
Dividing in standard form
Evaluate (4.5 × 106) ÷ (9 × 10−2), giving your answer in standard form.
divide the a-parts, subtract the powers4.5 ÷ 9 = 0.5106 ÷ 10−2 = 106−(−2) = 1080.5 is not between 1 and 10 — rewrite it0.5 = 5 × 10−1 ⇒ 5 × 10−1 × 1085 × 107subtracting a negative power adds — and a small a-part drops the power by 1.
WE 5
Adding in standard form
Evaluate (5 × 107) + (8 × 106), giving your answer in standard form.
match the powers — use the higher one, 1078 × 106 = 0.8 × 107now add the a-parts(5 + 0.8) × 107 = 5.8 × 1075.8 × 107rewrite the smaller number to share the bigger power, then add.
WE 6
Full question: a space probe
A space probe is 4.5 × 109 km from Earth. Radio signals travel at 3 × 105 km/s. (a) Write the distance as an ordinary number. (b) Find the time, in seconds, for a signal to reach the probe, in standard form. (c) A second probe is 9 × 108 km from Earth — how much further away is the first probe? Give your answer in standard form.
(a) move the decimal 9 places right4.5 × 109 = 4 500 000 000 km(b) time = distance ÷ speed(4.5 ÷ 3) = 1.5 and 109 ÷ 105 = 104(c) subtract — match powers at 1099 × 108 = 0.9 × 109 ⇒ (4.5 − 0.9) × 109(a) 4 500 000 000 km · (b) 1.5 × 104 s · (c) 3.6 × 109 kmdivision: a-parts and powers separately; subtraction: match the powers first.
💡 Top tips
Check a every time — it must be 1 ≤ a < 10: a single non-zero digit before the decimal point.
The sign of n is a quick reality check: large number ⇒ positive, number below 1 ⇒ negative.
To multiply or divide, treat the a-parts and the powers of 10 as two separate calculations.
To add or subtract, match the powers first — never combine a-parts that sit on different powers of 10.
In scientific mode the GDC keeps answers in standard form — useful for checking, but show the steps on Paper 1.
âš Common mistakes
Leaving a outside 1–10 — answers like 42 × 1013 or 0.5 × 108 are not yet in standard form.
Wrong sign for n — a small number such as 0.0000078 needs a negative power.
Miscounting decimal jumps — recount carefully, especially with strings of zeros.
Adding a-parts on different powers — 5 × 107 + 8 × 106 is not 13 × 1013; match the powers first.
Next up: Approximation — rounding to significant figures and decimal places, and knowing when a context forces you to round up. The standard-form habit carries straight over: pin down the leading digits first, then decide the size of the number.
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