In this lesson we discuss the different kinds of number systems with a special emphasis on rational and irrational numbers. The slides go through what distinguishes a rational number from an irrational number, some common forms of each, and the best way to recognize and deal with these two number groups. You will see how to name each using place value and fraction sense and then convert between decimals and fractions.

So without further ado, read through the slides below to get a feel for what decimals and fractions are and how the two are interchangeable!

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

**Adding and Subtracting Fractions**

This lesson is a video generated from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. You may also head over to my YouTube channel to see the slideshows in video form there!

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

- rational numbers
- Fractions and Decimals
- Objectives
- By the end of this lesson you should feel comfortable:
- Recognizing fractions and decimals
- Converting between fractions and decimals
- Parts of Numbers
- As we discussed in the last lesson, parts of numbers can come in very handy for things such as:
- Money
- Averages
- Measurements
- Pieces
- Very rarely will these numbers turn out to be nice, whole numbers
- Notice the two types of rational numbers you see here:
- decimals
- A
**decimal**is a number that contains any part of a whole represented behind a period. - Observe our old friend the number line:
- As you can see, we now have decimals between our whole numbers.
- Between 0 and 1 we have 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9.
- But that isn’t all, we could have any part, no matter how small!
- Decimals
- You may have seen numbers like this spoken before:
- Perhaps you would say “four point three two five.”
- While this is a common way to state this decimal number, there is a more proper way using place value.
- You are hopefully familiar with the
*ones*place. - But what happens after the decimal?
- Imagine if you have one whole square. Then you cut it into 10 groups:
- Each row would be a
*tenth*of the square. - Decimals
- Therefore, we would call the first place after the decimal the
*tenths*place as it takes 10 to make one whole. - For the next place, let’s say we cut each of those rows into ten sections:
- Now we have 100 sections, making the second place value after the decimal the
*hundredths*place. - If we were to separate each of these sections into 10 more, we would have one thousand pieces, or a
*thousandths*place. - And this pattern would continue on for infinite places.
- Decimals
- Now that we know our place values, we can properly name our decimal number.
- To the left of the decimal we have four.
- The decimal itself is just read as “and.”
- To the right of the decimal we have three hundred twenty-five which extends into the
*thousandths*place. - Therefore our name must be:
- Fractions
- So if it is acceptable most times to just read off our number as “four point three two five,” then why do we have this more official and difficult naming system?
- Decimals are not the only way to represent parts.
- Fractions are another way to show rational numbers.
- Let’s take a look back to the beginning:
- What fraction of these circles are shaded black?
- 3 out of 6, which we read as three
*sixths.* - fractions
- Sixths is the
*denominator*of our fraction, which is like its name or the type of fraction we are dealing with. - Denominator is actually from the Latin
*denominare**,*which means “to name.” - I like to remember it as “the denominator is de name of de fraction.”
- Again, why all this talk about proper names?
- Notice the
*ths*on the word*sixths*? Looks a lot like our naming convention for decimals doesn’t it? - Decimals to Fractions
- Four and three hundred twenty-five thousandths.
*Thousandths*implies that our denominator would be 1000.- Therefore our fractional part would be:
- In front we still have our whole part, 4.
- Thus we can see that these are two equivalent forms of the same number.
- We can use either fractions or decimals to represent the same rational number!
- Decimals to Fractions
- In answer to our question about why mathematicians name the way they do, remember that we are lazy creatures!
- The proper name for a decimal or fraction very easily lets us convert from a decimal to a fraction or
*mixed number*, like you see here. - We can reduce our fraction, though.
- Now our fraction has the same value but is in the form
*4 and thirteen fortieths.* - In the second form, it is much more difficult to transform our number back into a decimal using names alone.
- Fractions to Decimals
- So then how would we go about changing a fraction into a decimal?
- In some cases it is easy such as:
- We have 7 as our whole, and 3
*tenths*as the part of a whole. - Thus 7 and 3 tenths would look like:
- But what about our difficult case from our previous example?
- We don’t see tenths, hundredths, or thousandths, so we will need to utilize some division.
- Fractions to Decimals
- Our fraction is thirteen fortieths, which means we have 13 out of forty parts.
- Or another way to think of this would be thirteen parts divided into 40 groups, because the fraction bar is another notation for division.
- Using our old friend long division, we can see that dividing 13 by 40 gives us the decimal 0.325.
- Along with our whole part, 4, in the ones place we have:
- And we know from earlier this is exactly the decimal we want!
- conclusion
- Both decimals and fractions are in the category of rational numbers.
**Decimals**are written as a whole with the part behind it separated by a decimal point.**Fractions**are written as a number over a number to represent a part of a whole.- Decimals can be named by place value starting with the
*tenths*place to the right of the decimal. - You can use this naming convention to convert decimals into fractions then reduce.
- To convert fractions with difficult denominators into decimals, you divide the numerator by the denominator.