IB Maths AI SLTopic 1 — Number & AlgebraPaper 1 & 2GDC-friendly~8 min read
Standard Form
Standard form (also called scientific notation) is a clean way to write numbers that are either huge (mass of the Sun) or tiny (size of an atom). The number is split into two parts: a coefficient a between 1 and 10, and a power of 10 that tells you which way and how far to shift the decimal point. Once you have everything in this form, multiplying, dividing, adding and subtracting follows simple index-law rules — and your GDC will happily handle the arithmetic for you.
📘 What you need to know
The form: every number can be written as a × 10n, where 1 ≤ a < 10 and n is an integer (positive, negative, or zero).
Positive n means a large number (decimal point moves right when expanding); negative n means a small number (decimal point moves left).
The coefficient a is strict: 1 ≤ a < 10 means values like 3.24 or 9.99 are valid; 10, 0.5, 12.4 are NOT — you must adjust the power of 10.
Multiplication rule: (a×10m) × (b×10n) = (a×b) × 10m+n. Multiply the coefficients, ADD the powers.
Division rule: (a×10m) ÷ (b×10n) = (a÷b) × 10m−n. Divide the coefficients, SUBTRACT the powers.
Addition/subtraction rule: first match the powers of 10 (use the HIGHER power), then add or subtract the coefficients.
Always re-check 1 ≤ a < 10 at the end: if your final coefficient is ≥ 10 or < 1, shift the decimal and adjust the exponent.
GDC will do it for you: set scientific (SCI) mode on your GDC and any answer pops out in standard form automatically.
The form — coefficient × power of ten
Standard form — formula bookleta × 10n, where 1 ≤ a < 10 and n ∈ ℤ
Count the places the decimal point has to move to land just after the first non-zero digit. Direction tells you the sign of n; the count of jumps is the size of n.
Operations in standard form
Multiplying & dividing
multiply/divide a‘s ADD/SUBTRACT powers
Combine coefficients with × or ÷; combine powers using xm·xn = xm+n or subtract for division.
Adding & subtracting
match powers FIRST, then combine
Convert both numbers so they share the same (higher) power of 10. Then add or subtract the coefficients; the power stays the same.
Always finish with a check: is your final coefficient in the range 1 ≤ a < 10? If it’s 22.8 or 0.5, shift the decimal one place and adjust the exponent by ±1. This single check catches the most common exam slip.
🧭 Recipe — putting a number into standard form
Find a: place the decimal point so exactly one non-zero digit sits to its left. This is your a; it will automatically satisfy 1 ≤ a < 10.
Count the jumps: count how many places the decimal point moved from its ORIGINAL position to its NEW position. This gives the magnitude of n.
Decide the sign of n: if the original number was large (≥ 10), n is positive. If it was small (< 1), n is negative. If it was already between 1 and 10, n = 0.
Combine: write the number as a × 10n. For operations, apply the index laws (add powers when multiplying, subtract when dividing).
Re-check the range: after any calculation, ensure 1 ≤ a < 10 still holds. If not, shift the decimal one place and adjust n by ±1.
Worked examples
WE 1
Large number — astronomical context
The average distance from Earth to the Sun is 149 600 000 km. Write this number in standard form.
Step 1 — find a (decimal point goes after the first non-zero digit)149 600 000. → 1.496…a = 1.496 ✓ (1 ≤ 1.496 < 10)Step 2 — count the jumpsdecimal point moved LEFT by 8 places→ n = +8 (large number, positive)Step 3 — write in standard form1.496 × 10⁸ kmsanity check: 1.496 × 10⁸ = 1.496 × 100 000 000 = 149 600 000 ✓. Most science calculations involving distance to the Sun use this exact value — it’s so common it has a name (1 AU, one astronomical unit).
WE 2
Small number — negative exponent
A typical virus has a diameter of 0.000 000 072 5 metres. Write this in standard form.
Step 1 — find a0.000 000 072 5 → 7.25…a = 7.25 ✓ (1 ≤ 7.25 < 10)Step 2 — count the jumpsdecimal point moved RIGHT by 8 places→ n = −8 (small number, negative)Step 3 — write in standard form7.25 × 10⁻⁸ mcheck by expanding: 7.25 × 10⁻⁸ = 7.25 ÷ 10⁸ = 7.25 ÷ 100 000 000 = 0.000 000 072 5 ✓. Notice the count of jumps (8) is one more than the count of zeros immediately after the decimal (7) — common slip!
WE 3
Multiplication — coefficient needs adjustment
Calculate (5 × 10⁷) × (8 × 10⁴), giving your answer in standard form.
Step 1 — multiply the coefficients, ADD the powerscoefficients: 5 × 8 = 40powers: 10⁷ × 10⁴ = 10⁷⁺⁴ = 10¹¹combined: 40 × 10¹¹Step 2 — check 1 ≤ a < 10a = 40 is NOT in range! (40 ≥ 10)Step 3 — adjust: write 40 in standard form first40 = 4.0 × 10¹so 40 × 10¹¹ = 4 × 10¹ × 10¹¹ = 4 × 10¹⁺¹¹4 × 10¹²on a GDC in SCI mode this is automatic — but in Paper 1 (no GDC for parts of it) and for showing method, always write the intermediate “40 = 4 × 10¹” step. Without that adjustment “40 × 10¹¹” technically isn’t standard form.
WE 4
Division — coefficient less than 1
Calculate (3 × 10⁹) ÷ (6 × 10⁻²), giving your answer in standard form.
Step 1 — divide coefficients, SUBTRACT powerscoefficients: 3 ÷ 6 = 0.5powers: 10⁹ ÷ 10⁻² = 10⁹⁻⁽⁻²⁾ = 10⁹⁺² = 10¹¹combined: 0.5 × 10¹¹Step 2 — check 1 ≤ a < 10a = 0.5 is NOT in range! (0.5 < 1)Step 3 — adjust: shift decimal RIGHT by 1, reduce exponent by 10.5 = 5 × 10⁻¹so 0.5 × 10¹¹ = 5 × 10⁻¹ × 10¹¹ = 5 × 10⁻¹⁺¹¹5 × 10¹⁰key sign trick: when subtracting a NEGATIVE exponent, double-negate to plus. 9 − (−2) = 9 + 2 = 11. This is the most common arithmetic slip — many candidates write “9 − 2 = 7” by mistake.
WE 5
Addition with different (negative) exponents
Calculate (6 × 10⁻⁵) + (4 × 10⁻⁶), giving your answer in standard form.
Step 1 — identify the HIGHER power of 10−5 > −6, so 10⁻⁵ is higher(remember: less negative = higher)Step 2 — match powers: convert 4 × 10⁻⁶ to base 10⁻⁵4 × 10⁻⁶ = 0.4 × 10⁻⁵(decimal shifted LEFT by 1, exponent went UP by 1)Step 3 — add the coefficients (powers stay the same)(6 + 0.4) × 10⁻⁵ = 6.4 × 10⁻⁵Step 4 — check 1 ≤ a < 106.4 is in range ✓6.4 × 10⁻⁵cross-check by expanding: 6×10⁻⁵ = 0.000 06, 4×10⁻⁶ = 0.000 004, sum = 0.000 064 = 6.4×10⁻⁵ ✓. With negative exponents, “higher” means less negative — −5 is higher than −20, not lower.
WE 6
Word problem — total mass calculation
A petri dish contains 2.4 × 10⁷ bacteria. Each bacterium has a mass of 9.5 × 10⁻¹³ kg. Find the total mass of bacteria in the petri dish, giving your answer in standard form.
Step 1 — set up: total mass = number × mass per bacteriumM = (2.4 × 10⁷) × (9.5 × 10⁻¹³)Step 2 — multiply coefficients, ADD powers2.4 × 9.5 = 22.810⁷ × 10⁻¹³ = 10⁷⁺⁽⁻¹³⁾ = 10⁻⁶combined: 22.8 × 10⁻⁶Step 3 — adjust: 22.8 ≥ 10, so shift22.8 = 2.28 × 10¹22.8 × 10⁻⁶ = 2.28 × 10¹ × 10⁻⁶ = 2.28 × 10¹⁺⁽⁻⁶⁾M = 2.28 × 10⁻⁵ kga tiny number — about 23 millionths of a kg, or 23 micrograms. Sanity check: a HUGE count (tens of millions) times a TINY mass should give something small but non-negligible — answer feels right. Standard form makes word problems like this much cleaner than chasing zeros.
💡 Top tips
Set your GDC to SCI mode: in scientific mode, every answer is automatically given in standard form. The display might look like “4.2E5” or “4.2×10⁵” depending on your model — both mean 4.2 × 10⁵.
Always re-check 1 ≤ a < 10 at the end: this is the single most common reason candidates lose marks. After ANY operation, look at your coefficient and ask “is it between 1 and 10?”.
Sign of n = direction of jump: jump LEFT (number gets smaller, so original was bigger) → n positive. Jump RIGHT (number gets bigger, so original was smaller) → n negative.
For addition/subtraction, pick the HIGHER power: it’s easier to convert downward (making the coefficient smaller, like 4 → 0.4) than upward.
Show the “40 = 4 × 10¹” adjustment step in your working — examiners want to see you noticed and fixed the out-of-range coefficient. Just writing the final answer can lose the method mark.
⚠ Common mistakes
Coefficient ≥ 10 or < 1 in the final answer: writing 40 × 10¹¹ or 0.5 × 10⁻³ is NOT standard form. You MUST adjust to bring the coefficient into the range 1 ≤ a < 10.
Wrong sign for n: confusing large and small numbers. Quick check: a positive n should expand to a number ≥ 10; a negative n should expand to a number < 1. If 5 × 10⁻³ looks like a large number to you, re-check.
Subtracting negative powers wrongly: 10⁹ ÷ 10⁻² = 10⁹⁻⁽⁻²⁾ = 10¹¹, NOT 10⁷. The minus-times-minus turns the subtraction into addition. Slow down with the signs.
Adding/subtracting without matching powers: you CANNOT do (6 × 10⁻⁵) + (4 × 10⁻⁶) by just adding 6 + 4 and keeping the power. The powers must match first, then combine coefficients.
Counting zeros instead of decimal jumps: for 0.000 072 5, the exponent is −5 (5 jumps to put the decimal after 7), NOT −4 (the number of zeros immediately after the decimal point). Always count the JUMPS the decimal makes.
Up next: Laws of Indices. Standard form rests on one big idea — adding and subtracting powers of 10. Index laws generalise that to ANY base: xm·xn = xm+n, (xm)n = xmn, and the zero/negative cases. They’re the toolkit you’ll use everywhere from exponential models to compound interest.
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