Number - Decimals and Place Value

The Place Value System: The decimal system is an extension of the base-10 system. To the left of the decimal point, values increase by multiples of $10$ (Ones, Tens, Hundreds). To the right, values decrease by divisors of $10$: Tenths ($0.1$ or $\frac{1}{10}$), Hundredths ($0.01$ or $\frac{1}{100}$), and Thousandths ($0.001$ or $\frac{1}{1000}$). Visually, imagine a place value chart where the decimal point is a fixed anchor; moving one column to the right is equivalent to dividing a block into $10$ smaller, equal pieces.

Expanded Form: Decimals can be broken down into the sum of the values of their individual digits. For example, $5.238$ is written as $5 + 0.2 + 0.03 + 0.008$. Visually, this is like taking a number and 'stretching' it out to see exactly how many wholes, tenths, and hundredths it contains.

Equivalent Decimals and Placeholders: Adding zeros to the end of a decimal (trailing zeros) does not change its value. For example, $0.6 = 0.60 = 0.600$. Visually, if you have a $10 \times 10$ grid representing one whole, shading $6$ full columns (6 tenths) covers the exact same area as shading $60$ small individual squares (60 hundredths).

Comparing and Ordering: To compare decimals, align the decimal points vertically so that digits in the same place value column can be compared directly. If the numbers have different lengths, use placeholder zeros to make them the same length. Visually, this aligns the 'units' with 'units' and 'tenths' with 'tenths', similar to lining up two rulers to see which is longer.

Rounding Decimals: To round a decimal to a specific place, look at the digit to its immediate right. If that digit is $5$ or higher, round up; if it is $4$ or lower, keep the digit the same. Visually, imagine a number line: if a number is at or past the halfway mark between two points, it 'rolls' forward to the larger value.

Multiplying and Dividing by Powers of 10: Multiplying by $10, 100, \text{ or } 1000$ moves the decimal point to the right by the number of zeros in the multiplier. Dividing moves the decimal point to the left. Visually, this is like the digits 'sliding' across the place value columns while the decimal point stays in the same spot.

Arithmetic with Decimals: For addition and subtraction, decimal points must be aligned vertically to ensure you are adding like parts (e.g., tenths to tenths). For multiplication, multiply the numbers as if they were whole numbers, then place the decimal point so the answer has the same total number of decimal places as the original factors combined.