You may already know that decimals are those little dots between numbers that look like periods. But why are they important, and what do they do? Keep reading to learn all about decimals and decimal place values in small and large numbers.
What Is a Decimal?
The decimal system, also known as the base 10 system, represents fractions and mixed numbers in decimal notation. A decimal point separates a whole number from the parts of the next number. The word decimal comes from the Latin word decimalis, which means "tenth."
Examples of numbers in decimal notation include:
In each of these examples, the placement of the decimal tells us how big (or small) the number is. A small decimal in the incorrect place can lead to a major error!
Decimal Place Value
Decimal place value refers to the place value of numbers smaller than one. You may already be familiar with place value in larger numbers, but it works the same way in very small numbers as well.
For example, in the number 1.23456, the place value looks like this:
- 1 - ones place
- 2 - tenths place
- 3 - hundredths place
- 4 - thousandths place
- 5 - ten thousandths place
- 6 - hundred thousandths place
Moving the decimal point to the right can make the number larger by a power of ten. For example:
- 1.23456 x 10 = 12.3456
- 12.3456 x 10 = 123.456
- 123.456 x 10 = 1234.56
If you move the decimal point to the left, you're making the number smaller by a power of ten. For example:
- 1.23456 ÷ 10 = 0.123456
- 0.123456 ÷ 10 = 0.0123456
- 0.0123456 ÷ 10 = 0.00123456
Understanding how the decimal impacts the size of the number can help you conceptualize multiplication much more easily. It can also help you identify whether a number is very large or very small, depending on the placement of the decimal.
Rounding With Decimals
The process of rounding with decimals is the same as rounding with whole numbers. If the digit you're looking at is a 1, 2, 3, or 4, the digit on its left stays the same (rounds down). If it's 5 or higher, the digit on its left goes up by one digit (rounds up).
For example, if you round up to one decimal place or the nearest one:
- 2.9 rounds up to 3 (the 9 tenths rounds up to 1, and 2 + 1 = 3)
- 89.2 rounds down to 89 (the 2 tenths rounds down to 0, and 89 + 0 = 89)
- 0.5 rounds up to 1 (the 5 tenths rounds up to 1, and 0 + 1 = 1)
- 100.4 rounds down to 100 (the 4 tenths rounds down to 0, and 100 + 0 = 100)
The same goes when you round to two decimal places, or round to the nearest tenth.
- 2.87 rounds up to 2.9 (7 hundredths rounds up to 1 tenth, and 2.8 + 0.1 = 2.9)
- 89.21 rounds down to 89.2 (1 hundredth rounds down to 0 tenths, and 89.2 + 0.0 = 89.2)
- 0.55 rounds up to 0.6 (5 hundredths rounds up to 1 tenth, and 0.5 + 0.1 = 0.6)
- 100.43 rounds down to 100.4 (3 hundredths rounds down to 0 tenths, and 100.4 + 0.0 = 100.4
If you round to three decimal places (or round to the nearest one hundredth), it looks like this:
- 2.871 rounds down to 2.87 (1 thousandth rounds down to zero hundredths, and 2.87 + 0.00 = 2.87)
- 89.205 rounds up to 89.21 (5 thousandths rounds up to one hundredth, and 89.20 + 0.01 = 89.21)
- 0.553 rounds down to 0.55 (3 thousandths rounds down to zero hundredths, and 0.55 + 0.00 = 0.55)
- 100.429 rounds up to 100.43 (9 thousandths rounds up to one hundredth, and 100.42 + 0.01 = 100.43)
The rounding rule applies whether you round to four decimal places (the nearest one-thousandth), five decimal places (the nearest ten-thousandth), six decimal places (the nearest hundred-thousandth), or lower. Because the numbers to the right of a decimal point are parts of the next number, rounding up provides some of the missing pieces to that number.
Decimals vs. Fractions
Decimals are essentially fractions written in a linear way. They both include number parts and whole numbers. Their structure is only different in the following way:
- Fractions put the part (numerator) over the whole (denominator), separated by a fraction bar — for example, ½ or ¾
- Decimals put the part after the whole, separated by a decimal point — for example, 0.5 or 0.75
The numbers ½ and 0.5 mean the same thing: one half. The fraction shows that there is one part present out of two, and the decimal shows that five parts are present out of 10. They are just different ways to express the same concept.
Decimals in Scientific Notation
Decimals operate within scientific notation to express very large or very small numbers in mathematical shorthand. When you have a large number, you use decimals and exponents to show it as a power of ten. The steps to converting numbers to scientific notation using decimals are:
- Look at the number. (85,000)
- Move the decimal to the left until it passes the first non-zero integer. (8.5000)
- The number of times you moved the decimal is the power of the exponent. (You moved the decimal four times from 85,000.0 to 8.5, so it's 104)
- Make the new decimal the coefficient, and multiply it by the exponent. (8.5. x 104)
- Now it's in scientific notation!
When working with a smaller decimal, it works the opposite way. The decimal moves to the right until it passes the first non-zero integer, and the exponent is a negative power of ten. (For example, 0.00085 becomes (8.5. x 10-4). That negative exponent demonstrates the difference between 85,000 and 0.00085, so make sure it's right!
The Importance of Proper Decimals
An improper decimal point can make a world of difference. After all, if 1,000 is ten times greater than 100, it's one thousand times greater than 1 — and ten thousand times greater than 0.1! Practice your decimal skills with a list of metric system prefixes that rely on the decimal notation system. Then, learn about the other types of numbers. You can also sharpen your math basics with examples of the main parts of a subtraction problem.