Sometimes my students struggle a bit with rounding when given a certain amount of significant figures. So let's have a look at how it works – especially some tricky bits with zeros all around.

A brief theory about significant figures: it means we talk about non-zero digits. If I look at a number, the first non-zero digit from the left side is my first significant figure. Then, depending on how many significant figures I want to have, that's how many digits I'll be working with. Just be careful and don't lose any important zeros at the end (1902 rounded to three significant figures is 1900, because think about it this way: if you were to say you have roughly £1900 or £190 or even £19, that's a very significant difference, right? =)

When talking about rounding to 1, 2, 3 – or as many as you want – significant figures, we could say that we have two main groups: numbers bigger than 1 and numbers smaller than 1.

Numbers Bigger Than 1:

Simply, the first non-zero number is "our" number. Take as many digits as needed and round the last one correctly. Be careful about zeros that come after the significant digits.

Example: 45,963 - three significant figures. The significant figures are 4, 5, 9. The next digit is 6, so we round up the last significant figure (9 becomes 10, carrying over). Therefore, 45,963 rounds to 46,000. The first three digits are significant; the other two hold the place value. So we actually had to write five digits altogether.

Tricky one: 678 - four significant figures. Since 678 only has three significant figures, to express it with four, you would conceptually add a zero after the decimal point if it were to imply precision: 678.0. (If more significant figures are needed beyond existing digits, you can add trailing zeros after the decimal point to indicate that precision).

Numbers Smaller Than 1:

The first non-zero number is what we are looking for, and then again, count the rest as required. Be careful about zeros at the beginning (and behind as well). Place values still matter!

Example: 0.45963 — four significant figures. The first non-zero digit is 4, so that's the first significant figure. We need four: 4, 5, 9, 6. The next digit is 3, so we round down (the 6 stays as is). Therefore, 0.45963 rounds to 0.4596.

Tricky one: 0.0006010045 - five significant figures. The first non-zero digit is 6. We need five significant figures: 6, 0, 1, 0, 0. The next digit is 4, so we round down (the last 0 stays as is). Therefore, 0.0006010045 rounds to 0.00060100.

Not good enough? Here are a few more examples; then you'll become experts in rounding numbers to any amount of significant figures...

One significant figure:

• 469,304 -> 500,000 (4 is the first s.f., next is 6, so round up)
• 469,304.7089 -> 500,000 (same logic, we couldn't care less about the decimals here)
• 1.09 -> 1 (1 is the first s.f., next is 0, so round down)
• 1.9 -> 2 (1 is the first s.f., next is 9, so round up)
• 0.00506798 -> 0.005 (5 is the first s.f., next is 0, so round down)
• 0.004801 -> 0.005 (4 is the first s.f., next is 8, so round up)
• 0.1 -> 0.1 (1 is the first and only significant figure)
• 0.0001 -> 0.0001 (same logic)

These last two might look confusing but don't forget we need to find the first non-zero digit, which defines the start of significant figures. That's why both last numbers are still rounded to one significant figure only, even though the last one has way more digits than the one before it. Maths is fun, right?

Two significant figures:

• 6982 -> 7000 (6, 9 are significant, next is 8, so round up the 9 to 10, carrying over)
• 6.982 -> 7.0 (6, 9 are significant, next is 8, so round up. We add .0 to show two significant figures)
• 0.1 -> 0.10 (1 is the first s.f., need two, so add a trailing zero to indicate precision – yes, we really need to add one more zero to have one more figure behind the first one)

Three significant figures:

• 9708 -> 9710 (9, 7, 0 are significant, next is 8, so round up the 0)
• 0.9708 -> 0.971 (9, 7, 0 are significant, next is 8, so round up the 0)

Four significant figures:

• 1000.09585 -> 1000 (the first 1 and the three zeros are significant; next is 0, so no rounding up)
• 100.09585 -> 100.1 (1, 0, 0, 0 are significant. Next is 9, so round up the last 0)
• 10.09585 -> 10.10 (1, 0, 0, 9 are significant; next is 5, so round up the 9, making it 10 behind the decimal point)
• 1.09585 -> 1.096 (1, 0, 9, 5 are significant; next is 8, so round up the 5)
• 0.19585 -> 0.1959 (1, 9, 5, 8 are significant; next is 5, so round up the 8)

Five significant figures:

• 101.109 -> 101.11 (1, 0, 1, 1, 0 are significant; next is 9, so round up the 0)
• 10.1109 -> 10.111 (same logic)
• 1.01109 -> 1.0111 (same logic)
• 0.101109 -> 0.10111 (same logic)

But of course, for the last one, we have to maintain the zero before the decimal point, so it looks like we have 6 digits. You may have noticed in many examples above that this happens sometimes. And even weirder things can happen; things like 400.1 rounded to 1, 2 and even 3 significant figures will give us in all three cases the same answer: 400. And that's ok, that's completely normal for rounding with significant figures.

A bit of madness in otherwise very logical maths, right?

At the end, maybe it's worth noting that rounding to decimal places is different from rounding to significant figures. Decimal places count digits to the right of the decimal point only. Significant figures can freely cross the decimal point from left to right – depending on the first non-zero digit. So 1.23456 rounded to 2 significant figures is 1.2, but rounded to two decimal places is 1.23.

So – have you mastered it already?