1.7 Decimals

Understanding Decimal Numbers

Decimal numbers let you represent parts of a whole using the base-10 place value system. They show up constantly in money, measurements, and scientific data, so getting comfortable with them now makes everything from fractions to percentages easier to work with.

Decimal Numbers in Context

Each digit to the right of the decimal point represents a fractional part of a power of 10:

• Tenths (first digit right of the decimal): $\frac{1}{10}$. So 0.3 means three tenths.
• Hundredths (second digit): $\frac{1}{100}$. So 0.07 means seven hundredths.
• Thousandths (third digit): $\frac{1}{1000}$. So 0.002 means two thousandths.

The pattern continues: each position is worth one-tenth of the position to its left, just like the whole-number side of the decimal point.

Decimals can be written in different forms:

• Standard form: 3.14
• Word form: "three and fourteen hundredths" (the word "and" marks where the decimal point goes)
• Expanded form: $3 + \frac{1}{10} + \frac{4}{100}$

You'll see decimals in context all the time. $1.50 means one dollar and fifty cents. A measurement of 1.75 meters means one meter and seventy-five centimeters.

Rounding Techniques for Decimals

Rounding decimals follows a simple process:

1. Identify the place value you're rounding to (tenths, hundredths, etc.).
2. Look at the digit one place to the right of that position.
3. If that digit is 5 or greater, increase the rounding digit by 1. If it's less than 5, keep the rounding digit the same.
4. Drop all digits to the right of the rounding position.

For example, 3.27 rounded to the nearest tenth: the digit after the tenths place is 7 (which is ≥ 5), so round up to 3.3. And 3.42 rounded to the nearest tenth: the digit after tenths is 2 (which is < 5), so it stays 3.4.

Rounding is also useful for estimation. If you need to quickly check whether an answer makes sense, round first: $3.14 + 2.72$ rounds to roughly $3 + 3 = 6$. If your calculated answer came out to 0.586, something went wrong.

Performing Operations with Decimals

Addition and Subtraction of Decimals

The key rule: line up the decimal points vertically. Then add or subtract column by column, just like with whole numbers.

1. Write the numbers so the decimal points are directly above each other.
2. Fill in empty places with zeros so both numbers have the same number of decimal digits. For example, $3.14 + 0.7$ becomes $3.14 + 0.70$.
3. Add or subtract starting from the rightmost column, carrying or borrowing as needed.
4. Place the decimal point in your answer directly below the other decimal points.

$3.14 + 0.70 = 3.84$

Multiplication and Division of Decimals

Multiplication works differently from addition. You don't need to line up decimal points. Instead:

1. Ignore the decimal points and multiply as if both numbers were whole numbers. For $3.2 \times 1.5$, multiply $32 \times 15 = 480$.
2. Count the total number of decimal places in both original factors. 3.2 has 1 decimal place, 1.5 has 1 decimal place, so the total is 2.
3. Place the decimal point in your product so it has that many decimal places: 480 becomes 4.80 (or 4.8).

Division requires you to eliminate the decimal in the divisor first:

1. Move the decimal point in the divisor to the right until it becomes a whole number. For $1.2 \div 0.4$, move the decimal in 0.4 one place right to get 4.
2. Move the decimal point in the dividend the same number of places. 1.2 becomes 12.
3. Divide normally: $12 \div 4 = 3$.

The reason this works is that you're multiplying both numbers by the same power of 10, which doesn't change the quotient.

Conversion Between Numerical Forms

Decimals to fractions:

1. Write the decimal over the appropriate power of 10. For 0.3, that's $\frac{3}{10}$. For 0.07, that's $\frac{7}{100}$.
2. Simplify if possible. $\frac{3}{10}$ is already in simplest form (3 and 10 share no common factor other than 1). $\frac{7}{100}$ is also already simplified.

Fractions to decimals:

Divide the numerator by the denominator. $\frac{1}{4} = 1 \div 4 = 0.25$. $\frac{3}{8} = 3 \div 8 = 0.375$.

Decimals to percentages:

Move the decimal point two places to the right and add a percent sign. This is the same as multiplying by 100.

• $0.75 = 75\%$
• $0.08 = 8\%$

Percentages to decimals:

Do the reverse. Remove the percent sign and move the decimal two places to the left.

• $45\% = 0.45$
• $7\% = 0.07$

Advanced Decimal Concepts

• Order of operations applies to decimals the same way it applies to whole numbers (PEMDAS/BODMAS). Decimals don't change the rules.
• Scientific notation uses decimals and powers of 10 to express very large or very small numbers, like $3.0 \times 10^8$ for the speed of light in meters per second.
• When working with measurements, the precision of your final answer is limited by the least precise measurement you started with.