Comprehensive Guide to Decimals
1. Introduction to Decimals
Decimal notation uses base-ten place values. Every terminating decimal can be written as a fraction with a denominator that is a power of 10. Non-terminating decimals use infinitely many decimal places. The decimal point separates the whole number part from the fractional part.
Examples:
- 3.75 = 3 + 7/10 + 5/100 = 3 + 0.7 + 0.05
- 0.125 = 1/10 + 2/100 + 5/1000 = 0.1 + 0.02 + 0.005
- 10.5 = 10 + 5/10 = 10 + 0.5
2. Types of Decimals
2.1 Terminating Decimals
These are decimal numbers that end after a certain number of digits.
Examples:
- 0.25 (ends after 2 decimal places)
- 3.75 (ends after 2 decimal places)
- 0.375 (ends after 3 decimal places)
2.2 Repeating Decimals
These are decimal numbers where a digit or sequence of digits repeats infinitely.
Examples:
- 0.333333... = 0.3 (the digit 3 repeats infinitely)
- 0.272727... = 0.27 (the sequence '27' repeats infinitely)
- 1.414141... = 1.41 (the sequence '41' repeats infinitely)
2.3 Non-terminating, Non-repeating Decimals
These decimals continue infinitely without an eventually repeating block. They represent irrational numbers, such as π and √2; rational numbers have terminating or eventually repeating decimal expansions.
Examples:
- π = 3.141592653589...
- √2 = 1.41421356237...
- e = 2.71828182845...
3. Converting Between Fractions and Decimals
3.1 Fraction to Decimal
Method: Divide the numerator by the denominator.
Example: Convert 3/4 to a decimal
3 ÷ 4 = 0.75
Example: Convert 2/3 to a decimal
2 ÷ 3 = 0.666666... = 0.6
3.2 Decimal to Fraction
For Terminating Decimals:
Method: Write as a fraction with the denominator based on the place value of the last digit, then simplify.
Example: Convert 0.75 to a fraction
0.75 = 75/100 = 3/4 (after simplifying)
For Repeating Decimals:
Method 1: Algebraic approach for simple repeating patterns.
Example: Convert 0.3 to a fraction
Let x = 0.333333...
10x = 3.333333...
10x - x = 3.333333... - 0.333333...
9x = 3
x = 3/9 = 1/3
Method 2: For more complex repeating patterns.
Example: Convert 0.27 to a fraction
Let x = 0.272727...
100x = 27.2727...
100x - x = 27.2727... - 0.2727...
99x = 27
x = 27/99 = 3/11
4. Operations with Decimals
4.1 Addition and Subtraction
Method: Align the decimal points and add or subtract as with whole numbers.
Example: 3.75 + 2.8
3.75 + 2.80 ------ 6.55
Example: 5.63 - 2.45
5.63 - 2.45 ------ 3.18
4.2 Multiplication
Method: Multiply as with whole numbers, then place the decimal point by counting the total number of decimal places in the factors.
Example: 2.3 × 1.4
23 ×14 --- 92 (23 × 4) 230 (23 × 10) --- 322 2.3 and 1.4 each have one decimal place. Therefore 2.3 × 1.4 = 322 ÷ 100 = 3.22.
4.3 Division
Method: Convert the divisor to a whole number by moving the decimal point, then move the decimal point in the dividend the same number of places. Divide the resulting numbers, keeping the decimal place value in the quotient. The divisor must be nonzero.
Example: 4.2 ÷ 0.6
First, convert 0.6 to 6 by moving the decimal point one place right.
Do the same with 4.2, becoming 42.
Now divide: 42 ÷ 6 = 7
Example: 3.75 ÷ 1.5
Convert to 37.5 ÷ 15
37.5 ÷ 15 = 2.5
5. Rounding Decimals
The steps below use the usual school rule for non-negative numbers: a tie rounds up. Other tie-breaking conventions exist. For negative numbers, specify the convention; with ties away from zero, -1.25 rounds to -1.3 to the nearest tenth.
Method:
- Identify the place value to which you want to round.
- Look at the digit one place to the right of that place value.
- If it's less than 5, round down (keep the digit in the rounding place the same).
- If it's 5 or greater, round up (increase the digit in the rounding place by 1).
Example: Round 3.748 to the nearest tenth
The digit in the tenths place is 7.
The digit to its right is 4, which is less than 5.
So we round down: 3.748 rounds to 3.7
Example: Round 5.286 to the nearest hundredth
The digit in the hundredths place is 8.
The digit to its right is 6, which is 5 or greater.
So we round up: 5.286 rounds to 5.29
6. Comparing Decimals
The place-value steps below compare non-negative decimals. Every negative number is less than zero or a positive number. For two negative numbers, the one with the smaller absolute value is greater: -2.7 < -2.07.
Method:
- Compare the whole number parts first.
- If they're equal, compare the tenths places.
- If those are equal, compare the hundredths places, and so on.
- For easier comparison, you can append zeros after the last fractional digit to make the numbers have the same number of decimal places.
Example: Compare 3.42 and 3.4
Make them have the same number of decimal places: 3.42 and 3.40
The whole numbers (3) are the same.
The tenths (4) are the same.
The hundredths: 2 > 0
Therefore, 3.42 > 3.4
Example: Compare 2.7 and 2.07
Make them have the same number of decimal places: 2.70 and 2.07
The whole numbers (2) are the same.
The tenths: 7 > 0
Therefore, 2.7 > 2.07
7. Decimals and Percentages
Decimal to Percentage: Multiply by 100 and add a % symbol.
Examples:
- 0.25 = 0.25 × 100% = 25%
- 0.075 = 0.075 × 100% = 7.5%
- 1.5 = 1.5 × 100% = 150%
Percentage to Decimal: Divide by 100 (remove the % symbol and move the decimal point two places to the left).
Examples:
- 45% = 45 ÷ 100 = 0.45
- 7.5% = 7.5 ÷ 100 = 0.075
- 250% = 250 ÷ 100 = 2.5
8. Word Problems with Decimals
Problem: Janet bought a shirt for $24.99, pants for $35.50, and socks for $8.75. How much did she spend in total?
Solution:
Total cost = $24.99 + $35.50 + $8.75
24.99 35.50 + 8.75 ------ 69.24
Janet spent $69.24 in total.
Problem: A board is 2.5 meters long. If you cut off 0.75 meters, how much is left?
Solution:
Remaining length = 2.5 - 0.75
2.50 - 0.75 ------ 1.75
There is 1.75 meters of board remaining.
Problem: A recipe requires 2.25 cups of flour. If you want to make 3 batches, how much flour will you need?
Solution:
Total flour needed = 2.25 × 3
2.25 × 3 ------ 6.75
You will need 6.75 cups of flour.
Problem: A 2.5 kg bag of rice costs $8.75. What is the cost per kilogram?
Solution:
Cost per kilogram = Total cost ÷ Total weight
Cost per kilogram = $8.75 ÷ 2.5
Converting to division by a whole number: $87.5 ÷ 25 = $3.5
The rice costs $3.50 per kilogram.
9. Interactive Decimal Quiz
Test Your Decimal Knowledge
Answer the following questions to test your understanding of decimals. Include % for question 5. Question 10 requires exact repeating notation, not a rounded finite decimal. You can edit answers and check again; Reset Quiz clears the results.
