Decimals

Introduction

Decimals are a key concept in mathematics that represent numbers that are not whole. They are used to express parts of a whole number or a fraction in a more concise form. In this topic, we will explore the basics of decimals, including how to read, write, compare, and perform operations with them.

Place Value

Decimals are based on the place value system, just like whole numbers. Each digit in a decimal number has a place value, which helps determine its position and significance. The place values to the right of the decimal point are powers of 10, with the first digit after the decimal point representing tenths, the second representing hundredths, and so on.

Example:

Given the decimal number $5.732$:

• The digit $5$ is in the units place.
• The digit $7$ is in the tenths place.
• The digit $3$ is in the hundredths place.
• The digit $2$ is in the thousandths place.

Reading and Writing Decimals

To read a decimal number, start from the left and say the whole number part. Then, say "point" before reading the digits to the right of the decimal point individually. For writing decimals, always place the decimal point in the correct position according to the place value of each digit.

Example:

Read and write the decimal $6.125$.

• Reading: Six point one two five.
• Writing: $6.125$

Comparing Decimals

When comparing decimals, start from the left and compare the whole number parts. If they are equal, move to the first decimal place and compare the digits in the tenths place. Continue this process until you find the first pair of digits that are not equal.

Example:

Compare $3.45$ and $3.456$.

• $3.45 < 3.456$ as $5 < 6$.

Adding and Subtracting Decimals

When adding or subtracting decimals, line up the decimal points and then perform the operation as you would with whole numbers. Remember to carry over any excess values to the next place value.

Example:

Calculate $2.34 + 1.56$. $$ \begin{array}{r} & 2.34 \

• & 1.56 \ \hline & 3.90 \ \end{array} $$

Multiplying Decimals

To multiply decimals, ignore the decimal points and multiply the numbers as if they were whole numbers. Then, count the total number of decimal places in the factors and place the decimal point in the product to match that total.

Example:

Find the product of $2.1 \times 3.5$. $$ \begin{array}{r} & 2.1 \ \times & 3.5 \ \hline & 7.35 \ & 6.3\phantom{0} \ \hline & 7.35 \ \end{array} $$

Dividing Decimals

When dividing decimals, move the decimal point in the divisor to make it a whole number. Then, move the decimal point in the dividend by the same number of places. Perform the division as you would with whole numbers.

Example:

Divide $4.32 \div 0.6$. $$ \begin{array}{r} & 4.32 \ \div & 0.6 \ \hline & 7.2 \ \end{array} $$

Common Mistakes

• Forgetting to line up decimal points when adding or subtracting decimals.
• Misplacing the decimal point in the answer when multiplying or dividing decimals.
• Incorrectly comparing decimals without considering all the digits.

Key Points

• Decimals represent numbers that are not whole.
• Place value is crucial in understanding the significance of each digit in a decimal number.
• Reading and writing decimals involve stating the whole number part and the parts to the right of the decimal point.
• When adding or subtracting decimals, line up the decimal points before performing the operation.
• Multiplying decimals involves ignoring the decimal points initially and adjusting the decimal point in the product.
• Dividing decimals requires moving the decimal point in both the dividend and divisor.

Practice Questions

1. Calculate $8.73 + 4.25$.
2. Compare $5.67$ and $5.6701$. Which is greater?
3. Find the product of $3.2 \times 0.5$.
4. Divide $9.6 \div 0.8$.
5. Solve $12.45 - 3.78$.

Practice Questions Solutions

1. $8.73 + 4.25 = 12.98$
2. $5.67 < 5.6701$
3. $3.2 \times 0.5 = 1.6$
4. $9.6 \div 0.8 = 12$
5. $12.45 - 3.78 = 8.67$