Understanding Decimals: Parts of a Whole
Hello Mathematicians! Welcome to the Decimals chapter, a foundational part of "Numbers and the number system." Don't worry if this chapter has felt tricky before. Decimals are simply another way of representing numbers that are not whole—they represent parts of a whole, just like fractions do!
You use decimals every single day: when dealing with money (\(\$1.50\)), measurements (a race time of \(9.8\) seconds), or weights. Mastering them is essential for success in IGCSE Mathematics.
1. Place Value: Knowing Your Positions
The most important rule in decimals is Place Value. The position of a digit tells you its size.
The decimal point (\(.\)) is the dividing line. Everything to the left is a whole number (Units, Tens, Hundreds). Everything to the right is a fraction of one.
Understanding the Fractional Places
As you move to the right of the decimal point, the value decreases by a factor of 10 each time:
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First place after the point: Tenths (\(1/10\) or \(10^{-1}\)).
Example: In \(0.7\), the 7 is 7 tenths.
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Second place after the point: Hundredths (\(1/100\) or \(10^{-2}\)).
Example: In \(0.04\), the 4 is 4 hundredths.
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Third place after the point: Thousandths (\(1/1000\) or \(10^{-3}\)).
Example: In \(0.002\), the 2 is 2 thousandths.
Memory Aid: Think of money. 1 whole dollar is the Unit. A dime is a Tenth (\(0.1\)). A penny is a Hundredth (\(0.01\)).
Key Takeaway: When reading a decimal like \(3.45\), you say "three and forty-five hundredths," because the last digit (5) is in the hundredths position.
2. Comparing and Ordering Decimals
Comparing decimals can be tricky because your brain wants to look at the numbers as whole integers.
Common Mistake to Avoid: Thinking that \(0.4\) is smaller than \(0.15\). (It looks like \(4\) vs \(15\), but that’s wrong!)
The Comparison Strategy: Padding with Zeros
To compare decimals easily, always make sure they all have the same number of decimal places by adding zeros to the end (this is called "padding"). Adding zeros to the right of the last digit does not change the value.
Example: Order these numbers from smallest to largest: \(0.5\), \(0.125\), \(0.48\)
- Find the longest decimal: \(0.125\) has three decimal places.
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Pad all others to three decimal places:
\(0.5\) becomes \(0.500\) (500 thousandths)
\(0.125\) stays \(0.125\) (125 thousandths)
\(0.48\) becomes \(0.480\) (480 thousandths) - Compare the new numbers: \(125 < 480 < 500\).
- Final Order: \(0.125\), \(0.48\), \(0.5\)
Quick Review: When comparing, always start with the digit furthest to the left (the highest place value).
3. Conversion Between Decimals and Fractions
Decimals and fractions are two sides of the same coin. You must be able to switch between them.
A. Decimal to Fraction
This is where place value shines! Use the name of the last place as your denominator.
Step-by-Step:
- Write the number over a power of 10: Count the number of digits after the decimal point. That tells you the number of zeros in your denominator (10, 100, 1000, etc.).
- Simplify (crucial step!): Divide the numerator and denominator by their highest common factor (HCF).
Example 1: Convert \(0.75\) to a fraction.
Two digits after the point means the denominator is 100.
\(\frac{75}{100}\)
Simplify by dividing numerator and denominator by 25:
\(\frac{75 \div 25}{100 \div 25} = \mathbf{\frac{3}{4}}\)
Example 2: Convert \(0.008\) to a fraction.
Three digits after the point means the denominator is 1000.
\(\frac{8}{1000}\)
Simplify by dividing numerator and denominator by 8:
\(\frac{8 \div 8}{1000 \div 8} = \mathbf{\frac{1}{125}}\)
B. Fraction to Decimal
A fraction is simply a division problem: Numerator divided by Denominator (Top divided by Bottom).
Example: Convert \(\frac{3}{8}\) to a decimal.
You calculate \(3 \div 8\).
\(3 \div 8 = \mathbf{0.375}\)
Did You Know? Fractions that result in decimals that stop (like \(0.375\)) are called terminating decimals. Fractions that result in decimals that repeat forever (like \(\frac{1}{3} = 0.333...\)) are called recurring decimals.
Key Takeaway: Place value (tenths, hundredths) dictates the denominator when moving from decimal to fraction.
4. Operations with Decimals (+, -, x, ÷)
Performing calculations with decimals requires specific rules to ensure the decimal point ends up in the correct place.
A. Addition and Subtraction
The rule here is simple, but crucial: Line up the decimal points!
Think of it like lining up people by height—you must align the "baseline" (the decimal point) first. If you have whole numbers, place a decimal point at the end and add zeros if needed (e.g., \(5 = 5.00\)).
Example (Addition): Calculate \(12.5 + 0.38 + 7\)
Align and pad the numbers:
\(12.50\)
\(0.38\)
\(+\) \(7.00\)
\(\overline{\mathbf{20.08}}\)
Tip: If you forget to line up the points, your answer will be drastically wrong!
B. Multiplication
Multiplication is handled in two stages:
- Ignore the decimal points and multiply the numbers as if they were whole.
- Count the places: Count the total number of digits after the decimal point in all the numbers you multiplied.
- Place the point: Starting from the right of your answer, move the decimal point left by the total number of places counted in Step 2.
Example: Calculate \(0.04 \times 1.2\)
- Multiply \(4 \times 12 = 48\).
- Count the decimal places: \(0.04\) has 2 places. \(1.2\) has 1 place. Total places = \(2 + 1 = 3\).
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Starting with 48, move the point 3 places left. You need to add a zero placeholder.
\(48 \rightarrow 0.48 \rightarrow 0.048\).
Answer: \(\mathbf{0.048}\)
C. Division
Division is the trickiest operation, but we have a reliable strategy: Make the divisor a whole number.
The divisor is the number you are dividing by (the second number). We cannot easily divide if the divisor is a decimal.
Example: Calculate \(5.4 \div 0.06\)
- Identify the Divisor: \(0.06\)
- Move the Decimal: Move the decimal point in the divisor until it is a whole number. (In \(0.06\), move it 2 places right to get 6).
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Do the Same to the Dividend: You must move the decimal point in the dividend (the first number, 5.4) by the exact same amount (2 places right).
\(5.4 \rightarrow 54.0 \rightarrow 540\) (We add a zero placeholder). - Perform the new division: The problem is now \(540 \div 6\).
\(540 \div 6 = \mathbf{90}\)
Analogy: Multiplying both the divisor and dividend by the same factor (like 10 or 100) is like multiplying the top and bottom of a fraction by the same number—it doesn't change the overall value!
Key Takeaway: For addition/subtraction, line up. For multiplication, count places. For division, clear the decimal in the divisor first.
5. Rounding Decimals
In exams, you will often be asked to round your answer. This is done in two main ways: rounding to a specific Decimal Place (DP) or rounding to Significant Figures (SF).
The Rule of the Decider: Look at the digit immediately after the position you are rounding to.
- If the decider is 5 or more (5, 6, 7, 8, 9), round up the previous digit.
- If the decider is 4 or less (0, 1, 2, 3, 4), keep the previous digit the same.
A. Rounding to Decimal Places (DP)
This means counting positions after the decimal point.
Example: Round \(3.14159\) to 3 Decimal Places.
- Find the 3rd decimal place: The digit 1 (Tenths, Hundredths, Thousandths).
- Look at the decider (the 4th digit): It is 5.
- Since 5 is 5 or more, round the previous digit (1) up to 2.
Answer: \(\mathbf{3.142}\)
B. Rounding to Significant Figures (SF)
Significant figures are the most important digits in a number.
How to find the 1st Significant Figure (SF):
- If the number is greater than 1, the 1st SF is the first digit from the left (e.g., in 72.8, the 7 is the 1st SF).
- If the number is less than 1, the 1st SF is the first non-zero digit after the decimal point (e.g., in 0.0059, the 5 is the 1st SF). Zeros at the beginning are only placeholders, not significant.
Example 1: Round \(0.04508\) to 3 Significant Figures.
- 1st SF is 4. 2nd SF is 5. 3rd SF is 0.
- The decider (the 4th SF) is 8.
- Since 8 is 5 or more, round the previous digit (0) up to 1.
Answer: \(\mathbf{0.0451}\) (The initial zeros must stay as placeholders).
Example 2: Round \(675,210\) to 2 Significant Figures.
- 1st SF is 6. 2nd SF is 7.
- The decider (the 3rd SF) is 5.
- Round 7 up to 8. Replace all subsequent digits with zeros to maintain the place value (675,210 is roughly 680,000).
Answer: \(\mathbf{680,000}\)
Key Takeaway: When rounding SF in whole numbers, use zeros to hold the place of the units, tens, etc., that you removed.
Chapter Summary: Decimal Superpowers
You now have the tools to handle decimals confidently! Remember these core rules:
Place Value: The position after the point (tenths, hundredths) defines the value.
Adding/Subtracting: Always line up the decimal point.
Multiplying: Count the total decimal places in the question to place the point in the answer.
Dividing: Make the divisor a whole number first!
Keep practicing these concepts, and you’ll find that decimals are totally manageable! Good luck!