1.3 Working With Decimals
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By the end of this section, you will be able to:
- interpret decimal notation and round decimals correctly
- add, subtract, and multiply decimals
- convert decimals to and from common fractions
- use decimals to calculate percents
Decimals are just another way we can express real numbers that are in between the whole numbers, and in fact they are basically shorthand for fractions whose denominators are powers of 10. See what I mean below...
Decimal | Fraction Equivalent | In Words |
"one-tenth" | ||
"one-hundredth" | ||
"one-thousandth" |
When we learn about whole numbers, we talk about place values in terms of powers of 10 as well. For a number like 1,357 we know that the 5 actually represents 5 tens, the 3 actually represents 3 hundreds, etc. Here's an example:
So you see we can think of the number in the image as listing out a bunch of components,
which can all be written in a single decimal representation of the value.
When I calculate final course grades for students, I generally round each number to the nearest integer, meaning if you got an 87.6, I just consider it an 88. In that case, I'm rounding to the ones place, but you can round to any place value you want. For example, when performing calculations about money, we round to two decimal places, because we only work with dollars and cents.
Rounding to a desired place value: If you want to round to the
Round
- nearest integer
- nearest hundred
- nearest tenth
- nearest hundredth
- nearest ten-thousandth (four decimal places)
- sixth decimal place
- Answer
-
- Find the ones digit (it's 1) and look to the right. We see a 7 next to it, which is bigger than 5, so we round up and give the answer as 202.
- Look at the hundreds digit (it's 2) and look to the right. We see a 0, which is less than 5, so we keep the 2 and write 200.
- Look at the tenths place (it's a 7). To the right, there is a 6, which is bigger than 5, so we round up to 201.8 as our answer.
- The hundredths place is the 6, and next to it we see a 5, so we round up and give 201.77 as our answer.
- The fourth decimal place is the 8, and next to it we see a 3, so we give 201.7658 as our answer.
- The sixth decimal place is the 2, and next to it is a 9, so we round up to 201.765833 as our answer.
Decimals are great, because in real life problems, the numbers don't generally stay cute and clean and whole. Decimals also, in my opinion, are way better to work with than mixed numbers, while giving a similar sort of intuition about the size of a quantity. However, I don't generally expect students to work with decimals much without using a calculator. I want you to know how to do reasonably simple arithmetic in your head, so we'll go over it briefly, but in your science classes you'll use technology to perform calculations with decimals.
Arithmetic With Decimals
To add or subtract two decimals,
- Determine the sign of the answer (positive or negative),
- Write them vertically with the decimal points aligned, using placeholder 0s if necessary, then...
- Add or subtract down columns as usual.
For example, to add
To subtract
To multiply two decimals,
- Determine the sign of the answer (positive or negative),
- Multiply them as if they were whole numbers, ignoring the decimal points for the moment, then...
- The new decimal point position is the sum of the decimal places in the two factors.
This sounds weird, but all I want you to take away from this is common sense like the following examples:
Multiply.
Solution
1. I just want to double
2. Ignoring the decimal points,
3. I can think of this as wanting "one-and-a-half of a 30." So
There is also a special trick you can use when multiplying or dividing by a power of 10 that you should be aware of.
When multiplying a decimal by a power of 10,
Compute.
Solution
1. We multiply
2. We get
3. The factor
When dividing a decimal by a
Compute.
1.
2.
3.
Solution
1. You don't really have to know anything about dividing here actually, if you remember that
2. Again,
3. Here we need to move the point three places to the left, to get
Converting Between Fractions and Decimals
For your health and happiness in daily life and future math and science courses, you should have a grasp of the decimal versions of commonly used fractions. Your average welder (I'm told by my brother, who builds rocket pads) knows all of his powers-of-2 fractions by heart:
Fraction | Decimal |
Translate from decimal to fraction or vice versa, as appropriate.
- Answer
-
because that's "two one-hundredths" if you say it in words. or if you forget, think "two tenths," which is the fractions , and then simplify! I know, this one isn't on the list, but notice that it's just two copies of and proceed...
While we're here, let's expand a bit on that "0.3 repeating" situation for
This fun fact works the other direction too, meaning that if you have a repeating decimal, it can be written as a ratio of two integers! If you want to learn how to translate back and forth, check out the end of the exercises section.
If a decimal never stops and there is no repeating pattern to its digits continuing forever, that's an irrational number.
Decimals and Percents
You paid for a math course, but you're getting some Latin today for free. The ending "-cent" comes from the Latin centum, which means a hundred, so the word "percent" means "out of a hundred." If I say 85% of my students like my jokes, that means out of every hundred students I teach, 85 of them like my jokes. This proportion can also be written as a fraction,
Here are the basic types of percentage computation you might be expected to know how to do. You would probably use a calculator for this type of problem.
I had 32 students last semester, and I know 85% of my students like my jokes. How many students liked my jokes last semester?
Solution
I need to answer the question, "What is 85% of 32?" We already know how to translate "85%" to math, and the word "of" is usually code for multiplication. I calculate
I took a poll of my 32 students, and actually only 22 thought my jokes were funny. :( What percentage of my students like my jokes?
Solution
I know that
Okay, that's the gist on what you should be able to do with decimals! Take a stab at the exercises in the next section.