What’s significant about Significant Figures in IB Maths Applications and Interpretation
Written by Ian Lucas
I'll start with a question: to how many significant figures is the number 600?
What does “significant” mean?
Consider the number 315 - which digit can you change which would have the least effect on the size of the number? Clearly the 5: increase or decrease it by 1, and 315 changes by 1. But move the 3 up or down 1, and the number will change by 100. So, the 3 is more "significant" than the 5 – and the further left a digit is, the more significant its value.
Why do we need to round numbers anyway?
Because, strange to say, numbers can be too accurate! What is the population of this town - answer: 14,192. But, hold on - someone has just been born; or someone has just died; or moved away, so the answer is no longer true. Better to say "about 14000." Or consider this calculation: a rectangle has sides measured at 2.1cm and 5.3cm, so its area is 11.13cm2. But since the dimensions were measurements, they weren't accurate - the area could be as low as 2.05 × 5.25, or as high as 2.15 × 5.35, giving a range 10.7625 to 11.5025. 11.13 is just too accurate - again, better to say "about 11cm2." So, when we are dealing with numbers which were originally measurements, or sampling data, we can never say such numbers are totally accurate.
How do I round 5.1495 to 2SF?
Ignore everything beyond the third significant figure, that is, the 4. Since 4 is less than 5, it doesn't "push" the 2 up to a 3, so the answer is 5.1 Put simply, 5.1495 is nearer to 5.1 than it is to 5.2.
If the third digit was a 5 (5.1595) then the number is nearer to 5.2 - this is why a 5 will always push the previous number up. And the same is true, of course, when dealing with decimal places.
The only issue is with 9s. For example, to round 3.5961 to 3SF, we see that the 6 pushes the 9 up, so we must also increase the 5 to a 6 - again, think of the 59 in the middle becoming 60; so the answer is 3.60.
But why 3.60 rather than 3.6?
When you consider their accuracy, these two numbers are different. If you were to mark 3.60 on a scale, the numbers on either side would be 3.59 and 3.61 - thus 3.60 represents the range 3.595 to 3.605. Whereas the numbers either side of 3.6 would be 3.5 and 3.7, and so 3.6 represents the range 3.55 to 3.65 - a much wider range of numbers.
If 2.7 is accurate to 2SF, is its upper bound 2.74 or 2.75?
In theory, 2.75 would be rounded up to 2.8 to 2SF, so the upper bound of 2.7 is actually 2.74999999999...... So in practice, when dealing with bounds, we use 2.75 as the upper bound.
How would I round 2317 to 3SF?
Beware! - the answer isn't 232. When you round a number, you should end up with a less accurate version of the number; but 232 is nowhere near 2317! Again, think in terms of "about" - 2317 is about 2320. We have to insert a 0 as a "place filler" to keep the number to the correct size.
So what about that 600 - how many SF?
The answer is that you can't tell! The zeroes may be significant (ie they have actual value 0), or they may be place fillers. So 600 could be accurate to 1, 2 or 3 SF! For example:
There are exactly 600 tea bags in this box.
All the zeroes are significant, and so the number is accurate to 3SF.
The coffee in this jar weighs 604 gm, but is labelled at 600 gm.
The first zero is significant, the second is a place filler. The number is accurate to 2SF.
The Battle of Agincourt was in 1415, about 600 years ago.
In fact, 605 years ago when I wrote this, and possibly more when you read it! So the number is only correct to 1SF. The two zeroes are both place fillers.
More useful insights into IB Maths Applications and Interpretation
Ian Lucas is the author of Peak’s study guide series for IB Maths Applications and Interpretation. His guides contain a thorough review of the syllabus topics as well as lots of practice problems and worked examples to ensure you understand everything from significant figures to probability. You can purchase his study guide for IB Maths Applications and Interpretation for Standard Level or Higher Level through our online store.
You can also check out Ian’s article Understanding conditional probability in IB Maths Applications and Interpretation on our IB Maths Applications and Interpretation subject resource page.
