Helpful Tools for Calculus, Chapter 0: Numbers, Notations and Notes
Previous Section: 0.4 Absolute Values and Intervals
These notes will often switch between fractions and decimals. For readers who are not comfortable with both forms of writing numbers, this section will be helpful to identify the ways one may be better than the other and how to convert between them.
Fractions
A fraction is generally written as 
which can be thought of the number of times that b can go into a. Some may choose to write fractions as 
which all mean the exact same thing but can often be confusing.
Fractions are particularly helpful for calculations, identifying patterns, and converting between units. We will discuss each of these below.
In terms of calculations, fractions are particularly helpful because multiplying, dividing, and exponentiating fractions is quite easy compared to decimals. This is because the rules for operations on fractions can often be split up and done independently to the top and bottom of the fraction, so that you are only using whole numbers. This will be expanded on in the next section.
Fractions are also very useful in identifying patterns. As an example, in Sequences and Series, a lot of time is dedicated to identifying patterns between coefficients. A sequence could be written in the following ways:
With decimals, the pattern can be really hard to identify! What’s the relationship between 
and 0.125? However, with fractions we can quickly see that the pattern is 
and that the number is increasing by 1.
In terms of converting between units, fractions are very helpful because both parts of the fraction (numerator/denominator or top/bottom) can represent a type of unit. So, if we wanted to find the number of hours in a week, only need to think about what middle steps might be important:
It is really helpful to see what units there are and which ones cancel out in many problems. This is called Dimensional Analysis using conversion factors.
Decimals
A decimal is more similar to a ‘normal’ number, but often represents parts instead of wholes. When using decimals, it is helpful to know place values very well:
0.12345… can be thought of as 1 tenth, 2 hundredths, 3 thousandths, 4 ten-thousandths, 5 hundred-thousandths, as so on. These names arise from the number of times we would need to multiply it to get a whole number ( so 0.02 is 2 hundredths and 0.02 x 100 = 2).
Additionally, scientific notation is particularly helpful for rounding or calculating with decimals. In scientific notation we prefer to have one number on the left of the decimal, so we want to write 0.12345 as 1.2345 and in order to do that we need to describe how many times the decimal needs to be moved. In this example, we moved the decimal once:
The exponent on 10 is the number of times the decimal is moved. It is negative if we moved the decimal right and positive if we moved it left. Here are some more examples:
Decimals are very helpful when trying to compare things using number sense. We can tell very quickly that 0.45 is larger than 
but if we needed to compare the fraction forms of these, 
, it is much harder to identify.
The other benefit to decimals is that it is easier to add and subtract them because we can treat them like larger numbers and just add the similar place values. With fractions, adding and subtracting is actually difficult sometimes.
Converting between Fractions and Decimals
The fraction a/b is just another way of saying a divided by b. This means that converting a fraction into a decimal can be done using division. For example, 4/5 can be divided the following way:
Similarly, 1/7 can be converted to a decimal like this:
…
At this point it should be clear that the decimal values will repeat, so we put a bar over the top of everything that is repeating, which is basically all of the terms!
To convert from a fraction to a decimal we write out the number, without the decimal, and divide it by the largest decimal place values and simplify as much as possible. For example:
Repeating decimals are a bit more challenging. To turn a repeating decimal into a fraction we need to do some algebra. Write the decimal and name it something, like N. Then above that write the same decimal with the point moved the same number of times as there are repeating terms. Multiply the other side by 10 for each time you do this. Then, you subtract the equations from each other and solve for N.
Review
Fractions and decimals are both representations of numbers, whole or partial. Fractions are specifically helpful for dimensional analysis and pattern solving while decimals are great for number sense and comparisons.
Long division is used to convert a fraction to a decimal
Using place values is used to convert a decimal to a fraction, when the decimal is terminating