Significant digits can be a frustrating concept if you do not understand the rules. Significant digits are the digits needed to have a precise answer. If you were to take a chemistry class, then you will be familiar with them. Recently, I began working with significant digits in my math class. I think the more you work on a concept the more you understand it. There are different rules when dealing with significant digits. Adding and subtracting numbers have different rules compared to multiplying and dividing numbers.
Adding and subtracting numbers focuses on the accuracy of the answer. The number with the least amount of decimal places is the “winning” number. For example if we are adding 5.36+2.2, the “winning” number is 2.2 because it has the least amount of decimal places. The answer to this problem is 7.56, but since we need to have 1 decimal place the answer is 7.6. In the picture below I have provided an example of a subtraction problem.
Multiplying and dividing numbers focuses on the least number of significant digits. Instead of looking at the amount of decimal places, we look at the whole number and count the amount of significant numbers; therefore, the “winning” number is the number with the least amount of significant digits. The solution to the problem should be rounded to the same amount of numbers as the “winning” number. For example if we are multiplying 82.1×3.2, the “winning” number is 3.2 because it only has 2 significant digits. In the picture above I have included a division problem and some tricky problems dealing with zeroes. I think with effort and determination anyone can understand the rules and differences between adding and subtracting numbers compared to multiplying and dividing!
Understanding the different types of numbers is great way to learn more about math in general. Real numbers are the main type of numbers that elementary students learn, but there are imaginary numbers that are learned during higher grade levels. Although real numbers are the main type of numbers, there are five different types of real numbers. Making a diagram of these different types is a great way to understand the differences.
There are also irrational numbers which are not in the diagram above. They are kind of in their own circle. These numbers are nonrepeating decimals that do not terminate and have no pattern. An example of an irrational number is the square root of 2. This number does not come to an end (not terminating) and has no pattern. If you type the square root of 2 into a calculator you will get something like 1.414213562 and it continues on and on. Clearly, you can see that this is an irrational number.
Adding and subtracting decimals are very similar concepts. In order to add and subtract numbers you need to line up the decimals in the problem. An example of adding two decimal numbers is 1.2+2.4. If we line up the decimal numbers it will be 3.6. Subtracting two decimal numbers like 3.6-2.1 uses the same method. You need to line up the decimals and subtract the numbers; therefore, the answer is 1.5. Lining up the decimal numbers can be hard if the numbers are super long. A great way to eliminate this problem is by lining up your decimal numbers on graph paper. I think this is a great tip that may create less confusion!
I think a great way to understand what decimals are is to put decimals into a table. As you can see from the table below, decimals can be written in many ways. One example of this is the number 0.1 can be written as 1 tenth, 10 raised to the -1, or 1/10. Decimals are base-ten units; therefore, we use the unit ten in the table.
The United States separates numbers with commas every third number: 5,853,567.75; however, many other places around the world separate their numbers with spaces and use a comma the way the United States uses the decimal point like this 5 853 597,75. I think this is a very interesting fact! Another thing I learned is that the number 0.0000000009 is easier to read if we put spaces every third number: 0.000 000 000 9. If we put spaces it does not mean anything different; it is just to help us count the zeros or read the number easier. These are great tips to keep in mind while working with decimals.
As you can see from the decimal table, decimals can be written in many ways; essentially a decimal is a fraction. I say this because 0.732 is 732/1000 and can be written as 732 thousandths. Another example of this is 0.20 is 20/100 and can be written as 20 hundredths. No matter the decimal number it can always be written as a fraction which may be easier for some people. You can also look at a decimal by expanding it. If we look back at the decimal 0.732 we can break it down into three parts 0.732=0.7+0.03+0.002. This can also be written as 732/1000=7/10+30/100+200/1000. Decimals are very versatile which allows for many different strategies to understand them better!