Fractions and Decimals - Decimal Place Value and Comparison

The Decimal Place Value Chart: A decimal number is divided into a whole number part and a fractional part by a decimal point. To the left of the decimal, we have units like Ones, Tens, and Hundreds. To the right, we have Tenths ($\frac{1}{10}$), Hundredths ($\frac{1}{100}$), and Thousandths ($\frac{1}{1000}$). As you move one position to the right, the value of the digit becomes 10 times smaller.

Understanding Tenths: The first digit to the right of the decimal point represents tenths. Visually, if you imagine a large square representing one whole unit divided into 10 equal vertical columns, each column represents $0.1$ or $\frac{1}{10}$.

Understanding Hundredths: The second digit to the right of the decimal point represents hundredths. Visually, if that same whole square is divided into a $10 \times 10$ grid of 100 small equal squares, each tiny square represents $0.01$ or $\frac{1}{100}$.

Understanding Thousandths: The third digit to the right of the decimal point represents thousandths. Visually, if a single hundredth square is further divided into 10 tiny slices, or if a large cube (one whole) is divided into $10 \times 10 \times 10$ small cubes, each small cube represents $0.001$ or $\frac{1}{1000}$.

Expanded Form of Decimals: This involves writing a decimal as a sum of the values of its individual digits. For example, the number $5.342$ is visualized as $5$ wholes, $3$ tenths, $4$ hundredths, and $2$ thousandths, written mathematically as $5 + 0.3 + 0.04 + 0.002$.

Comparing Decimals: To compare two decimals, align the decimal points and compare the digits starting from the largest place value (left to right). If the digits are the same, move to the next place value on the right. Using a number line helps visualize this; the number further to the right is always the greater value.

Equivalent Decimals and Placeholder Zeros: Adding zeros to the end of a decimal (to the right of the last digit) does not change its value. For example, $0.6 = 0.60 = 0.600$. This is helpful for 'padding' numbers so they have the same amount of digits when comparing values like $0.7$ and $0.65$ (making $0.7$ into $0.70$).

Decimals on a Number Line: Decimals can be plotted between whole numbers. For example, $0.5$ is exactly halfway between $0$ and $1$. Between $0.1$ and $0.2$, there are ten smaller intervals representing hundredths ($0.11, 0.12, \dots, 0.19$).