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Skill Checklist
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
8 Skills Available
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📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Rounding Numbers
2 skills
Rounding rules
SL Core 1.1
There are standardized rules for how to approximate numerical values:
If the digit after the one being rounded off is LESS than 5 (0,1,2,3, or 4), we round down.
If the digit after the one being rounded off is 5 OR MORE (5,6,7,8, or 9), we round up.
1.34∣5→1.351.34∣4→1.34
Significant figures
SL Core 1.1
A significant figure is any digit that is not a leading or trailing zero. To round off to a number of significant figures, count off the specified number of significant figures, then round based off of rounding rules (down if the next digit is less than 5, up if the digit is 5 or more).
NOTE: Unless a question states otherwise, on IB exams, you are expected to give answers to 3 significant figures.
Scientific Notation
3 skills
Writing numbers in standard form
SL Core 1.1
Scientific notation is a useful way to write large or small numbers in a compact form. It uses powers of 10 to "condense" a lot of digits. Numbers written in scientific notation are of the form
a×10k
where 1≤a<10 and k∈Z.
Scientific notation is sometimes called "standard form."
Adding & subtracting numbers in scientific notation
SL Core 1.1
Let x=3×105,y=4×106, and suppose we want to find x+y.
Since the powers of 10 are different, we cannot simply add 3+4. Instead, we rewrite y so it is multiplying 105:
y=4×106=4×105+1=40×105
Now we can add:
x+y=3×105+40×105=43×105
Finally, we convert back to scientific form, since 43>10:
x+y=4.3×106
Basically, we took the higher power of 10 and "split" it so that it matched the smaller power of 10.
Multiplying & dividing numbers in scientific notation
SL Core 1.1
We can multiply and divide numbers in scientific form as follows:
(3×109)×(4×105)=12×1014=1.2×1015
4×1053×109=0.75×104=7.5×103
Multiplying and dividing numbers in scientific notation relies heavily on exponent rules.
Error
3 skills
Approximation error bounds
SL AI 1.6
A measurement is always accurate to ±21 ("plus or minus one half") of the smallest division on the scale. The same idea applies to rounding numbers: the rounded value is accurate to ±21 the value of the digit "place" (tens, ones, tenths, hundredths, etc.) it is rounded to.
Typically, we express the error bounds of a number x using inequalities:
x−21u≤x<x+21u
where x is the measured value, u is the smallest unit on the scale we used. Note that some people equivalently write
x∈[x−21u,x+21u)
The upper bound, x+21u, is not included since it would be rounded up to the next increment.
Absolute and percentage error
SL AI 1.6
The actual size of an error is the difference between the approximated value (VA) and the exact value (VE). We call this the absolute error, and can calculate it with the equation
absolute error =∣VA−VE∣
Expressing the error as a percentage of the exact value allows us to see how much the error "matters" in a specific scenario. This is the percentage error, which we calculate with the equation
percentage error =VE∣VA−VE∣×100%
Bounds of calculation based on other approximations
SL AI 1.6
The error bounds of values like length and width impact the error bounds of any secondary calculation, like area, that we use them to find. For example, if an object of length l and width w has error bounds 0.9≤l≤1.1 and 2.6≤w≤2.9, then the object's minimum and maximum area are
Amin=0.9(2.6)=2.34
Amax=1.1(2.9)=3.19
i.e., the error bounds are 2.34≤A<3.19.