Understanding Decimals

Decimals are numbers that have a whole part and a fractional part, separated by a decimal point. In grade 4 math, learning about decimals includes understanding what they are, how to read and write them, and how to compare and operate on them. Decimals are an important part of math because they help us represent numbers that are not whole. For example, decimals are used to describe parts of a whole, such as dollars and cents in money, measurements, and more.

What is a decimal?

A decimal number is a number that includes a decimal point, which separates the whole number from the fractional part. For example, in the number 4.25, 4 is the whole number and 25 represents the fractional part. The decimal point is important because it tells us where the whole number ends and the fraction begins.

Place value in decimal

When working with decimals, it's important to understand place value. Every number in decimals has a place, or place value, just as in whole numbers. However, the places to the right of the decimal point have special names:

• The first place to the right of the decimal is the tenths place.
• The second place is the hundredth place.
• The third place is the thousandth place.

This is how place values work in decimal:

    whole decimal fraction
            point
     4 | . | 2 5
        unit | tenth | hundredth

In this example, 4.25, 4 is in the ones place, 2 is in the tenths place, and 5 is in the hundredths place.

Reading decimals

Reading decimals is fairly simple once you know the place value. There are two parts to read: the whole number part and the fractional part. You read the numbers on either side of the decimal point separately. For example:

• The decimal number 3.7 is read "three point seven."
• The decimal number 0.23 is read "zero point two three."
• The decimal number 12.456 is read "twelve point four five six."

Comparing decimals

To compare decimal numbers, you start at the left and move to the right, comparing each digit. The first number that is different tells you which decimal is larger:

• 0.7 and 0.6: Here the tenths place is different. 7 is greater than 6, so 0.7 is greater.
• 0.45 and 0.451: Both have the same 45 and hundredths, but the extra thousandth means that 0.451 is greater.

Operations with decimals

Adding and subtracting decimals

When adding or subtracting decimals, it's important to line up the decimal points. Let's see how this works with some examples:

Example 1: Addition

  4.26
+ 3.15
,
  7.41

Example 2: Subtraction

  5.50
- 2.25
,
  3.25

Multiplying decimals

Multiplying decimals involves multiplying them as if they were whole numbers, then calculating the total number of decimal places in each factor and placing the decimal in the product:

Example: 0.3 x 0.2 = ?

  0.3
x 0.2
,
  0.06 (3*2=6 and total decimal places=2, result=0.06)

Dividing decimals

When dividing a decimal by a whole number, divide as usual, and place the decimal point directly above its place in the dividend:

Example: 1.2 ÷ 2 = ?

 0.6
,
2 | 1.2

If you are dividing by a decimal, you must first make the divisor a whole number by moving the decimal place and making the same change to the dividend. This is done like this:

Example: 2.4 ÷ 0.6 = ?

 4
,
6 | 24

Common uses of decimal numbers

Decimals are used in many everyday situations:

1. Money: Dollars and cents are usually represented using decimal numbers. For example, $5.99 means 5 dollars and 99 cents.
2. Measurement: Length and weight are often measured in units that may require decimals. For example, 1.45kg means 1 kilogram and 450 grams.
3. Statistics: Decimals are used to represent results precisely, such as averages or probabilities like 0.75.

Converting fractions to decimals

To convert a fraction to a decimal, divide the numerator by the denominator. This can also help you understand the relationship between fractions and decimals.

Example: Convert 3/4 to a decimal.

  3 ÷ 4 = 0.75

Converting decimals to fractions

Converting a decimal to a fraction involves identifying the place value of the last digit of the decimal part and using it to form a fraction.

Example: Convert 0.25 to a fraction.

  0.25 = 25/100 = 1/4 after simplification

Practice problems

Let's solve some practice problems to ensure understanding:

1. Add 1.5 + 2.75
2. Subtract 3.4 - 1.2
3. Multiply 0.6 x 0.7
4. Divide 2.4 ÷ 0.8
5. Convert 7/10 to Decimal
6. Convert 0.5 to a fraction

Conclusion

Decimals are a versatile numerical system that allows us to perform a variety of calculations and express non-whole numbers succinctly. By understanding local values, learning how to perform operations with decimals, and knowing how to convert between decimals and fractions, we gain a powerful tool for everyday life and mathematical solutions.

These concepts are an important part of mathematical learning and application, laying the groundwork for more complex numerical skills in the future. As you practice and become more comfortable with decimals, you will find that working with them is more intuitive.

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