Lesson 1: Decimals

New Concept

Decimals are another way to write fractions with denominators that are powers of ten. This lesson covers how to name, write, convert, compare, and round them, providing a complete foundation for working with these essential numbers.

What’s next

Next, you'll explore interactive examples for naming and converting decimals, followed by practice cards to master locating and ordering them on the number line.

Property

To name a decimal number:

1. Name the number to the left of the decimal point (the whole number).
2. Write 'and' for the decimal point.
3. Name the number to the right of the decimal point as if it were a whole number.
4. Name the decimal place of the last digit. The 'th' at the end of the name means the number is a fraction.

Examples

• The number 4.3 is read as 'four and three tenths' because the 3 is in the tenths place.
• The number 2.45 is read as 'two and forty-five hundredths' because the last digit, 5, is in the hundredths place.
• The number 0.009 is read as 'nine thousandths'. We don't name the zero whole number.

Explanation

Naming decimals is like telling a number's full story. The part before 'and' is the whole number, and the part after is the fraction. The last word, like 'hundredths', tells you the size of the fractional pieces.

Property

To write a decimal number from its name:

1. Look for the word 'and'. It tells you where to place the decimal point.
2. Write the whole number to the left of the decimal point.
3. Identify the place value given by the last word (e.g., 'hundredths' means two decimal places).
4. Write the decimal part to the right of the decimal point, using zeros as placeholders if needed.

Examples

• To write 'fourteen and thirty-seven hundredths', the 'and' gives us 14.37, with two decimal places for 'hundredths'.
• To write 'twenty-four thousandths', there is no 'and', so we start with 0. 'Thousandths' means three places, so we write 0.024.
• 'Eight and three hundredths' is written as 8.03. We need a zero to hold the tenths place so the 3 is in the hundredths place.

Explanation

Think of 'and' as the signal to place your decimal point. The words before 'and' are the whole number, and the words after are the fraction part. The last word tells you how many decimal places to fill.

Property

To convert a decimal to a fraction or mixed number:

1. The number to the left of the decimal is the whole number part of the mixed number. If it is zero, you will have a proper fraction.
2. The number to the right of the decimal point becomes the numerator of the fraction.
3. The denominator is a power of 10 corresponding to the place value of the last digit.

Examples

• To convert 4.09, the whole number is 4. The numerator is 9 and the place value is hundredths, so it is $4\frac{9}{100}$.
• To convert 3.7, the whole number is 3. The numerator is 7 and the place value is tenths, resulting in $3\frac{7}{10}$.
• The decimal $-0.286$ has no whole number part. The fraction is $-\frac{286}{1000}$, which simplifies to $-\frac{143}{500}$.

Explanation

Converting a decimal is straightforward. The whole number stays the same. The digits after the decimal point go on top of the fraction, and the bottom is the place value of the very last digit (like 10, 100, or 1000).

Property

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line. To locate a decimal, first identify the integers it is between. Then, divide that interval into tenths, hundredths, or smaller parts to find the exact position.

Examples

• To locate 0.4, find the space between 0 and 1, divide it into 10 parts, and mark the fourth one. $0.4$ is equivalent to $\frac{4}{10}$.
• To locate $-0.74$, you look between 0 and $-1$. It is between $-0.70$ and $-0.80$, but a little closer to $-0.70$.
• To locate 3.1, you find the whole number 3 and then move one-tenth of the way toward 4.

Explanation

Think of a number line as a super detailed ruler. Find the whole numbers the decimal lives between. Then, zoom in and divide that space into 10 equal parts (tenths) to pinpoint its exact spot.

Property

To order decimals, compare the numbers from left to right. First, compare the whole number parts. If they are equal, compare the tenths place, then the hundredths place, and so on, until one digit is larger than the other. Two decimals are equivalent decimals if they convert to equivalent fractions; adding zeros to the end of a decimal does not change its value .

Examples

• To compare 0.64 and 0.6, write 0.6 as 0.60. Since 64 is greater than 60, we know that $0.64 > 0.6$.
• To compare 0.83 and 0.803, write 0.83 as 0.830. Since 830 is greater than 803, it follows that $0.83 > 0.803$.
• To order negative decimals like $-0.1$ and $-0.8$, remember that the number closer to zero is larger. Therefore, $-0.1 > -0.8$.

Explanation

To see which decimal is bigger, line them up by their decimal points. Pad them with zeros so they have the same length. Now, just compare the numbers as if the decimal point wasn't there!

Property

To round a decimal:

1. Locate the place value you are rounding to.
2. Look at the digit immediately to its right.
3. If that digit is 5 or greater, add 1 to the digit in the rounding place.
4. If that digit is less than 5, keep the digit in the rounding place the same.
5. Drop all digits to the right of the rounding place.

Examples

• To round 18.379 to the nearest hundredth, the digit to the right of the 7 is 9. Since $9 \ge 5$, we round up to get 18.38.
• To round 18.379 to the nearest tenth, the digit to the right of the 3 is 7. Since $7 \ge 5$, we round up to get 18.4.
• To round 18.379 to the nearest whole number, the digit to the right of the 8 is 3. Since $3 < 5$, the whole number stays 18.

Explanation

Rounding simplifies numbers. Find your target place, then peek at the digit next door to the right. If it's 5 or bigger, round your target up. If it's 4 or smaller, leave your target alone. Then, erase everything after it.