Standard Form
Hi! In this lesson we’re going to talk about standard form — also called scientific notation. It’s one of those tools that looks fancy but is actually a huge time-saver. By the end of this note, you’ll be writing massive numbers like the distance to the sun, and tiny numbers like the size of an atom, in just a few characters. Let’s go.
What you need to know
- Standard form means writing a number as a × 10ⁿ
- a must be between 1 and 10 (so 1 ≤ a < 10)
- n is positive for big numbers, n is negative for tiny numbers
- n tells you how many places the decimal moved — and in which direction
- Use it to multiply, divide, add, and subtract huge or tiny numbers easily
Why does standard form even exist?
Imagine I asked you to write down the mass of the Earth. It’s about 5 970 000 000 000 000 000 000 000 kilograms. Now imagine writing the mass of an electron — about 0.000 000 000 000 000 000 000 000 000 000 911 kilograms. Painful, right? You’d lose your mind counting zeros, and one little mistake throws off the whole answer.
This is exactly the problem standard form solves. Instead of writing all those zeros, we use the powers of 10 to do the heavy lifting:
Without standard form
Mass of Earth = 5 970 000 000 000 000 000 000 000 kg
Mass of electron = 0.000 000 000 000 000 000 000 000 000 000 911 kg
With standard form
Mass of Earth = 5.97 × 10²⁴ kg
Mass of electron = 9.11 × 10⁻³¹ kg
Think of standard form as a shorthand. The “a × 10ⁿ” format keeps the meaningful digits at the front, and lets the power of 10 handle all the zeros. Scientists, engineers, and your IB examiners love it because it’s compact, accurate, and impossible to miscount.
What is standard form?
The official definition is this: a number is in standard form when it’s written as:
Let me break that down piece by piece, because the rules sound boring but they really matter:
- a is the number at the front. The rule says 1 ≤ a < 10 — meaning a can be 1, 2.5, 9.99, but never 0.5 and never 10. It’s always a single digit before the decimal point (followed by any decimals you want).
- 10ⁿ is the power of 10. The little number “n” is called the exponent, and it can be a positive whole number, a negative whole number, or even zero.
- n is positive when the original number is bigger than 10 (like 240 or 5,000,000)
- n is negative when the original number is smaller than 1 (like 0.05 or 0.000003)
- n = 0 when the number is between 1 and 10 already (because 10⁰ = 1)
Quick reminder: the symbol ℤ just means “integer” — any whole number, positive, negative, or zero. So when an exam question says “give your answer in the form a × 10ⁿ where n ∈ ℤ”, they’re just saying “n must be a whole number”. No fractions allowed.
How do I write a big number in standard form?
Let’s take a real number and turn it into standard form together. Here’s our number:
The trick is to move the decimal point until you have one digit before it. Right now the decimal is hiding at the very end (3 240 000.). I want to move it left until I get a number between 1 and 10.
- Spot the decimal point. In a whole number like 3 240 000, the decimal point sits invisibly at the end → 3240000.
- Move the decimal left until “a” is between 1 and 10. I need to land between the 3 and the 2 → that gives me 3.240000, or just 3.24.
- Count how many places you moved. I shifted the decimal 6 places to the left. So n = 6.
- Write the answer: 3 240 000 = 3.24 × 10⁶
Let me sanity-check this. If I take 3.24 and multiply it by 10⁶ (which is 1,000,000), I get 3,240,000. ✓ Perfect.
Memory trick: for big numbers, the decimal moves LEFT and n is POSITIVE. Big number → big positive power.
How do I write a small number in standard form?
Now let’s flip it. What about a tiny number like:
Same idea, but this time we move the decimal point in the opposite direction.
- Find where the decimal currently is — right at the start: 0.000567.
- Move it RIGHT until “a” is between 1 and 10. I need to slide it past the zeros until I land just after the 5 → I get 5.67.
- Count the moves. I shifted the decimal 4 places to the right. Because I moved right (which makes the number bigger), I need a negative power of 10 to bring it back down. So n = −4.
- Write the answer: 0.000567 = 5.67 × 10⁻⁴
Check: 5.67 × 10⁻⁴ = 5.67 ÷ 10,000 = 0.000567. ✓ Spot on.
The pattern in one line: Decimal moves LEFT → n is POSITIVE. Decimal moves RIGHT → n is NEGATIVE. The number of moves IS the value of n.
Quick examples — try these in your head
Before we go further, let me show you a few numbers in both forms so the pattern locks in:
- 100 000 → the decimal moves 5 left → 1 × 10⁵
- 0.00001 → the decimal moves 5 right → 1 × 10⁻⁵
- 4 057.52 → the decimal moves 3 left → 4.05752 × 10³
- 0.00107 → the decimal moves 3 right → 1.07 × 10⁻³
- 7 → already between 1 and 10 → 7 × 10⁰ (remember 10⁰ = 1)
And going the other way — converting standard form back to a normal number:
- 4.501 × 10⁷ → move decimal 7 places to the right → 45 010 000
- 4.501 × 10⁻⁷ → move decimal 7 places to the left → 0.000 000 4501
How do I multiply numbers in standard form?
This is where standard form really pays off. Multiplying huge numbers becomes easy because you just handle the “a” parts and the “10ⁿ” parts separately.
The big idea: multiply the front numbers normally, and add the powers. That’s it. Two operations.
Let me show you with a clean example:
- Multiply the “a” parts: 3 × 4 = 12
- Add the powers: 10² × 10⁵ = 10²⁺⁵ = 10⁷
- Put it together: 12 × 10⁷
- BUT WAIT — 12 isn’t between 1 and 10, so this isn’t proper standard form. We need to fix it. Rewrite 12 as 1.2 × 10¹.
- So: 1.2 × 10¹ × 10⁷ = 1.2 × 10⁸ ✓
Now let’s try one with bigger numbers, where x = 3 × 10⁷ and y = 4 × 10⁹. Find xy.
- Multiply 3 × 4 = 12
- Add the powers: 10⁷ × 10⁹ = 10¹⁶
- Combine: 12 × 10¹⁶
- Fix “a”: 12 = 1.2 × 10¹, so the answer is 1.2 × 10¹⁷
How do I divide numbers in standard form?
Dividing is the same idea, but you subtract the powers instead of adding.
- Divide the “a” parts: 2 ÷ 8 = 0.25
- Subtract the powers: 10⁻³ ÷ 10⁻⁵ = 10⁻³⁻⁽⁻⁵⁾ = 10⁻³⁺⁵ = 10²
- Put it together: 0.25 × 10²
- Fix “a” — 0.25 isn’t between 1 and 10. Rewrite as 2.5 × 10⁻¹.
- Final answer: 2.5 × 10⁻¹ × 10² = 2.5 × 10¹ (which equals 25)
Watch the signs! When you “subtract a negative power,” it becomes positive. So 10⁻³ ÷ 10⁻⁵ = 10⁻³⁻⁽⁻⁵⁾ = 10⁻³⁺⁵ = 10². This is the most common place students lose marks — slow down here.
How do I add or subtract in standard form?
Adding and subtracting is a bit different — you can’t just add the powers. The rule is: both numbers must have the same power of 10 before you can add or subtract them.
Think of it like this: you can’t directly add 3 apples and 4 boxes of apples — you first have to express them in the same unit. Same here.
Example with positive powers
Same powers (easy case): if x = 3 × 10⁷ and y = 4 × 10⁷, then:
- Both have the same power (10⁷), so just add the “a” parts: 3 + 4 = 7
- Keep the same power: 10⁷
- Answer: x + y = 7 × 10⁷
Different powers (the trickier case): now x = 3 × 10⁷ and y = 4 × 10⁹. Find x + y:
- Pick the highest power. Here that’s 10⁹.
- Rewrite the smaller one to match. 3 × 10⁷ is 100 times smaller than something with 10⁹, so I write it as 0.03 × 10⁹.
- Now add normally: 0.03 × 10⁹ + 4 × 10⁹ = (0.03 + 4) × 10⁹ = 4.03 × 10⁹
- Final answer: 4.03 × 10⁹ ✓
Example with negative powers
It works exactly the same with negative powers — just remember: a more negative exponent means a smaller number. So 10⁻²⁰ is bigger than 10⁻²¹.
Find: (8 × 10⁻²⁰) − (5 × 10⁻²¹)
- Highest power = 10⁻²⁰ (because −20 is greater than −21)
- Rewrite 5 × 10⁻²¹ as 0.5 × 10⁻²⁰ (it’s 10 times smaller)
- Now subtract: (8 − 0.5) × 10⁻²⁰ = 7.5 × 10⁻²⁰
- Answer: 7.5 × 10⁻²⁰ ✓
Examiner Tips and Tricks
Use your GDC for the heavy lifting. Set your calculator to scientific mode and it will automatically display answers in standard form. This works for multiplication, division, addition, and subtraction — try inputting (3.6 × 10⁻³)(1.1 × 10⁻⁵) directly and watch the magic.
Watch out for E-notation! Your calculator might display “6.5E5” or “6.5×10^5” instead of 6.5 × 10⁵. They mean exactly the same thing — but in your final exam answer, always rewrite it cleanly as 6.5 × 10⁵.
3 significant figures is your friend. The IB usually wants final answers either in exact form or to 3 s.f. So 1.2345 × 10² becomes 1.23 × 10², and 1.2345 × 10⁻⁵ becomes 1.23 × 10⁻⁵. Don’t lose easy marks by giving too many or too few digits.
Worked Example
Calculate the following, giving your answer in the form a × 10ⁿ, where 1 ≤ a < 10 and n ∈ ℤ.
i) 3 780 × 200
Answer:
ii) (7 × 10⁵) − (5 × 10⁴)
Answer:
iii) (3.6 × 10⁻³)(1.1 × 10⁻⁵)
Answer:
iv) (4 × 10⁵⁰) + (2 × 10⁴⁸)
Answer:
Common Mistakes
- Leaving “a” outside 1 ≤ a < 10. Writing 12 × 10⁵ or 0.5 × 10³ is a guaranteed mark loss. Always check that “a” is a single digit before the decimal point.
- Adding/subtracting powers on plus or minus. The rule “add powers when multiplying” is for multiplication only — never for addition. Slow down and make the powers match first.
- Forgetting to subtract the powers when dividing. 10⁻³ ÷ 10⁻⁵ is NOT 10⁻⁸ — it’s 10². Subtract carefully, especially with negative signs.
- Confusing the direction of decimal jumps. Big number = decimal jumps LEFT, small number = decimal jumps RIGHT. Burn that into memory.
- Misreading E-notation. 6.5E5 means 6.5 × 10⁵, NOT 6.5 × 5. The E stands for “exponent of 10.”
- Mixing up “negative power = small number.” 10⁻²⁰ is BIGGER than 10⁻²¹, because −20 is greater than −21. This trips up almost everyone the first time.
Final word from your teacher: standard form looks intimidating because of all the powers and exponents, but it’s literally just decimal-shifting + a tiny bit of arithmetic. Once you’ve practiced 10–15 questions, it becomes muscle memory. Trust the steps, check your “a” is between 1 and 10, and you’ll fly through these in the exam.
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