1.4 Decimals

Decimals show up everywhere in algebra, from simplifying expressions to solving equations. This section covers decimal arithmetic, conversions between number forms, and how decimals connect to the real number system, including square roots and irrational numbers.

Decimal Basics

Rounding Decimals

Rounding a decimal means trimming it to a specific place value. Here's the process:

1. Identify the place value you're rounding to (tenths, hundredths, thousandths, etc.).
2. Look at the digit immediately to the right of that place.
3. If that digit is 5 or greater, increase the digit in your target place by 1 (round up).
4. If that digit is less than 5, leave the digit in your target place unchanged (round down).
5. Drop all digits to the right of the target place.

For example, rounding 3.4872 to the hundredths place: look at the digit in the thousandths place (7). Since 7 ≥ 5, round up the hundredths digit from 8 to 9. The result is 3.49.

Arithmetic with Decimals

Addition and subtraction:

1. Line up the decimal points vertically. Pad with zeros if the numbers have different lengths after the decimal.
2. Add or subtract as you would with whole numbers.
3. Place the decimal point in the result directly below the aligned decimal points.

For example, $3.7 + 12.45$ becomes $3.70 + 12.45 = 16.15$.

Multiplication:

1. Multiply the numbers as if they were whole numbers (ignore the decimals for now).
2. Count the total number of decimal places in both factors.
3. Place the decimal point in the product that many places from the right.

For example, $1.2 \times 0.03$: multiply $12 \times 3 = 36$. There are 1 + 2 = 3 total decimal places, so the answer is $0.036$.

Division:

1. If the divisor has a decimal, shift its decimal point to the right until it's a whole number.
2. Shift the dividend's decimal point the same number of places to the right.
3. Divide as you would with whole numbers.
4. Place the decimal point in the quotient directly above its new position in the dividend.

For example, $4.5 \div 0.3$: shift both decimals one place right to get $45 \div 3 = 15$.

Conversions Between Number Forms

Decimal to fraction:

1. Write the decimal over 1.
2. Multiply the numerator and denominator by $10$ for each digit after the decimal point.
3. Simplify the fraction.

For example, $2.5 = \frac{2.5}{1} = \frac{25}{10} = \frac{5}{2}$.

Fraction to decimal: Divide the numerator by the denominator. Some fractions terminate ($\frac{3}{8} = 0.375$), while others repeat ($\frac{1}{3} = 0.333...$). If the division doesn't terminate, round to the desired place value.

Decimal to percentage: Move the decimal point two places to the right and add a percent sign. $0.25 = 25\%$.

Percentage to decimal: Remove the percent sign and move the decimal point two places to the left. $40\% = 0.40$.

Real Numbers and Expressions

Simplifying Square Roots

A perfect square is a number whose square root is a whole number. The first several are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

• If the number under the radical is a perfect square, simplify directly: $\sqrt{25} = 5$
• If it's not a perfect square, factor out the largest perfect square factor:

1. Find the largest perfect square that divides evenly into the number.
2. Rewrite the radical as a product of two square roots.
3. Simplify the perfect square root.

For example: $\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$

Another example: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$. You could also factor as $\sqrt{4 \cdot 18}$, but pulling out the largest perfect square gets you to the simplified form in one step.

Classification in Real Numbers

The real number system is a set of nested categories, where each type contains the ones before it:

• Natural numbers: positive counting numbers starting from 1 (1, 2, 3, ...)
• Whole numbers: natural numbers plus 0 (0, 1, 2, 3, ...)
• Integers: whole numbers plus negatives (..., -3, -2, -1, 0, 1, 2, 3, ...)
• Rational numbers: any number that can be written as a fraction $\frac{a}{b}$, where $a$ is an integer and $b$ is a nonzero integer. This includes terminating decimals (0.5) and repeating decimals ($0.333... = \frac{1}{3}$).
• Irrational numbers: numbers that cannot be written as a fraction. Their decimal expansions go on forever without repeating. Examples: $\pi$, $\sqrt{2}$, $\sqrt{3}$.
• Real numbers include all rational and irrational numbers together.

A common point of confusion: every integer is also a rational number (for instance, $5 = \frac{5}{1}$). And every natural number is a whole number, an integer, a rational number, and a real number. The categories build on each other.

Number Line Representations

You can locate any real number on a number line by thinking in terms of place value.

• Fractions: Divide the unit interval into equal parts based on the denominator. For $\frac{3}{4}$, split the space between 0 and 1 into 4 equal parts and count 3 parts from 0.
• Decimals: Divide each unit interval into 10 equal parts for tenths, then each tenth into 10 parts for hundredths, and so on. For 0.75, go 7 tenths past 0, then 5 hundredths further.

This subdivision process reflects the base-10 structure of our number system. Every decimal, no matter how many digits, can be pinpointed on the number line this way.

Advanced Decimal Concepts

Scientific Notation and Significant Figures

Scientific notation expresses a number as a coefficient (between 1 and 10) multiplied by a power of 10. It's useful for very large or very small numbers.

For example: $602{,}000{,}000{,}000{,}000{,}000{,}000{,}000 = 6.02 \times 10^{23}$, and $0.00045 = 4.5 \times 10^{-4}$.

Significant figures indicate how precise a measurement is. The rules:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant (e.g., 305 has 3 significant figures).
• Leading zeros are not significant (e.g., 0.0042 has 2 significant figures).
• Trailing zeros after a decimal point are significant (e.g., 2.50 has 3 significant figures).

Decimal expansion refers to how a number is represented as digits after the decimal point. A decimal expansion can be terminating ($0.75$), repeating ($0.333...$), or non-terminating and non-repeating (which makes the number irrational).

Applications

Real-World Decimal Applications

When solving word problems involving decimals, follow these steps:

1. Identify the given information and what you need to find.
2. Determine which operation(s) to use (addition, subtraction, multiplication, division).
3. Perform the decimal arithmetic carefully.
4. Round your answer appropriately for the context. Money problems round to the hundredths place (cents). Measurements round to the precision given in the problem.
5. Check that your answer is reasonable. A sale price should be less than the original. A total should be more than any single item.

For example, if a shirt costs $24.99 and is 15% off, convert the percentage to a decimal ($0.15$), multiply ($24.99 \times 0.15 = 3.7485$), round to cents ($3.75), and subtract from the original price ($24.99 - 3.75 = 21.24$). The sale price is $21.24.