5.2 Decimal Scaling, Ordering & Negative Numbers

Multiplying and Dividing by Powers of 10: The Digit-Shift Principle

One of the most damaging misconceptions in primary mathematics is the rule: "To multiply by 10, just add a zero."

Consider what happens if you apply this rule to a decimal:

$$3.45 \times 10 \xrightarrow{\text{naive rule}} 3.450$$

Because trailing zeros at the end of a decimal do not alter its value, $3.450 = 3.45$. The number has not changed at all! The naive rule fails completely.

The Universal Rule: Digits Move, the Decimal Point Stays Fixed

The decimal point is an immovable anchor marking the boundary between whole numbers and fractions. When multiplying or dividing by powers of 10 ($10, 100, 1,000$), the digits shift across the columns:

1. Multiplying by $10$: Every digit shifts 1 column to the left (becoming 10 times larger).
2. Multiplying by $100$ ($10^2$): Every digit shifts 2 columns to the left (becoming 100 times larger).
3. Multiplying by $1,000$ ($10^3$): Every digit shifts 3 columns to the left (becoming 1,000 times larger).
4. Dividing by $10$: Every digit shifts 1 column to the right (becoming 10 times smaller).
5. Dividing by $100$ ($10^2$): Every digit shifts 2 columns to the right (becoming 100 times smaller).
6. Dividing by $1,000$ ($10^3$): Every digit shifts 3 columns to the right (becoming 1,000 times smaller).

If shifting digits leaves empty columns between a digit and the decimal point, place-holding zeros must be inserted.

Worked Example: Digit Shifts in Action

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Ordering and Comparing Integers and Decimals

Selective entrance assessments frequently present lists of decimals with differing lengths (ragged decimals) and ask candidates to arrange them in ascending (smallest to largest) or descending (largest to smallest) order.

The "Longer Decimals Are Larger" Trap

Young students often assume that because $45 > 4$, $0.045$ must be larger than $0.4$. This is completely false. The magnitude of a decimal is governed by the place value of its leading non-zero digits, not by the count of digits after the decimal point.

The 3-Step Decimal Ordering Protocol

To order any collection of decimals flawlessly under timed conditions, follow this three-step protocol:

[Step 1: Vertical Alignment]
Write all numbers in a vertical column with their decimal points strictly aligned.

[Step 2: Pad with Trailing Zeros]
Identify the number with the most decimal places and pad all other numbers with
trailing zeros so that every number has identical decimal length.

[Step 3: Systematic Column Comparison]
Compare digits column-by-column starting from the greatest place value (far left).
The first column with differing digits establishes the relative order.

Example Walkthrough

Arrange the following numbers in ascending order:

$$0.4, \quad 0.045, \quad 0.405, \quad 0.44, \quad 0.0405$$

• Step 1 & 2 (Pad to 4 decimal places):
  • $0.4000$
  • $0.0450$
  • $0.4050$
  • $0.4400$
  • $0.0405$
• Step 3 (Compare tenths, then hundredths, then thousandths):
  • Numbers with $0$ in the tenths column: $0.0405$ and $0.0450$. In thousandths, $0 < 5$, so $0.0405 < 0.045$.
  • Numbers with $4$ in the tenths column: $0.4000, 0.4050, 0.4400$.
    • Hundredths: $0.4000$ and $0.4050$ have $0$; $0.4400$ has $4$.
    • Thousandths: $0.4000$ has $0$; $0.4050$ has $5$, so $0.4000 < 0.4050 < 0.4400$.
• Final Ascending Order: $$0.0405 < 0.045 < 0.4 < 0.405 < 0.44$$

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Negative Numbers in Context & Number Line Navigation

Negative numbers represent quantities that fall below a zero reference point. In selective assessments, questions test negative numbers through three real-world contexts:

1. Temperature: Thermometers reading degrees Celsius above or below freezing ($0^\circ\text{C}$).
2. Elevation: Heights above sea level ($+8,848\text{ m}$) versus ocean trenches or land depressions below sea level ($-430\text{ m}$).
3. Financial Balances: Positive bank credits versus negative overdraft debts.

 ← Negative Direction (Subtracting / Falling)       Positive Direction (Adding / Rising) →
 ───┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬───
   -6    -5    -4    -3    -2    -1     0    +1    +2    +3    +4    +5    +6    +7

Number-line steps, with optional signed-number extensions

1. Adding a positive number: Move to the right on the number line ($-3 + 5 = +2$).

2. Subtracting a positive number: Move to the left on the number line ($2 - 6 = -4$). The following two algebraic rules extend the basic Year 5 work of interpreting negatives and counting through zero.

3. Adding a negative number: Adding debt reduces value; equivalent to subtraction ($a + (-b) = a - b$). For example: $4 + (-7) = 4 - 7 = -3$.

4. Subtracting a negative number: Removing debt increases value; equivalent to addition ($a - (-b) = a + b$). For example: $5 - (-3) = 5 + 3 = 8$.

Calculating Intervals Crossing Zero (Temperature & Elevation)

When calculating the temperature rise or total difference between a negative value and a positive value, never subtract the magnitudes naively:

$$\text{Distance between } -a \text{ and } +b = |-a| + b = a + b$$

• Worked Scenario: In Oslo, the midnight temperature was $-7^\circ\text{C}$. By 2:00 PM, the temperature rose to $12^\circ\text{C}$. What was the total temperature rise?
  • Distance from $-7^\circ\text{C}$ up to $0^\circ\text{C} = 7^\circ\text{C}$.
  • Distance from $0^\circ\text{C}$ up to $12^\circ\text{C} = 12^\circ\text{C}$.
  • Total rise: $7 + 12 = 19^\circ\text{C}$.
  • Algebraic check: $12 - (-7) = 12 + 7 = 19^\circ\text{C}$.

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Comprehensive Worked Examples

Example 1: Place Value and Decimal Shifts

Question: A scientist measures a microscopic organism at $0.035\text{ mm}$ long. In a high-powered digital microscope, the image is enlarged by a factor of $1,000$. What is the displayed length of the organism in centimetres?

Step-by-step Solution:

1. Calculate the enlarged length in millimetres by multiplying by $1,000$: $$0.035\text{ mm} \times 1,000$$ Shifting digits 3 places to the left: $0.035 \rightarrow 0.35 \rightarrow 3.5 \rightarrow 35\text{ mm}$.
2. Convert millimetres to centimetres. Since $1\text{ cm} = 10\text{ mm}$, divide by $10$: $$35\text{ mm} \div 10$$ Shifting digits 1 place to the right: $35 \rightarrow 3.5\text{ cm}$.
3. Final Answer: $3.5\text{ cm}$.

Example 2: Negative Number Intervals Across Zero

Question: A deep-sea research submarine descends from an elevation of $14\text{ m}$ above sea level to an underwater trench at $168\text{ m}$ below sea level. If the descent takes exactly $14$ minutes at a constant speed, how many metres does the submarine descend each minute?

Step-by-step Solution:

1. Calculate the total vertical distance descended: $$\text{Total distance} = 14\text{ m} - (-168\text{ m}) = 14 + 168 = 182\text{ m}$$
2. Divide the total distance by the time taken ($14\text{ minutes}$): $$182 \div 14$$ Using short division: $140 \div 14 = 10$, and $42 \div 14 = 3$. So $182 \div 14 = 13\text{ m/minute}$.
3. Final Answer: $13\text{ metres per minute}$.

Example 3: Multi-Digit Rounding with Constraints

Question: The attendance at a stadium championship match was reported as $68,000$ when rounded to the nearest thousand. What is the smallest possible number of spectators that could have been in attendance, and what is the largest possible number?

Step-by-step Solution:

1. For rounding to the nearest $1,000$, the decider column is the hundreds column.
2. Smallest possible integer (rounds UP to $68,000$): The lower midpoint is 500 below 68,000, so the smallest whole number rounding up to 68,000 is: $$67,500$$
3. Largest possible integer (rounds DOWN to $68,000$): The upper midpoint is 68,500, which rounds to 69,000. The largest whole number below that midpoint is: $$68,499$$
4. Final Answer: Minimum = $67,500$; Maximum = $68,499$.

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Section Review Checklist

• Place values shift by $\times 10$ (left) and $\div 10$ (right); decimal points do not move.
• Align ragged decimals vertically and pad with trailing zeros before comparing.
• Subtracting a negative number is equivalent to adding a positive ($a - (-b) = a + b$).
• Rounding decisions rely solely on the digit immediately to the right of the target place.
• Roman numerals use subtractive pairs only for 4, 9, 40, 90, 400, and 900.

1. Decimal Place Value & Fraction Equivalents

Terminating decimals can be written as fractions whose denominators are powers of ten ($10, 100, 1000$). The decimal point separates whole units from fractional parts:

• Tenths ($0.1$): Equal to $\frac{1}{10}$. One unit split into 10 equal parts.
• Hundredths ($0.01$): Equal to $\frac{1}{100}$. One unit split into 100 equal parts.
• Thousandths ($0.001$): Equal to $\frac{1}{1000}$. One unit split into 1000 equal parts.

Place Value Chart:
 Hundreds │ Tens │ Ones ┆ Tenths │ Hundredths │ Thousandths 
    100   │  10  │   1  ┆  0.1   │    0.01    │    0.001    
──────────┼──────┼──────┼────────┼────────────┼─────────────
     0    │   0  │   0  ┆   4    │      5     │      0      = 0.45
     0    │   0  │   0  ┆   1    │      2     │      5      = 0.125

Converting Terminating Decimals to Fractions

To convert a terminating decimal to a fraction in simplest form:

1. Count the number of decimal digits after the decimal point.
2. Write the digits as an integer numerator over a power of 10 ($10$ for 1 place, $100$ for 2 places, $1000$ for 3 places).
3. Simplify the resulting fraction by dividing by the Highest Common Factor (HCF).

Worked Example 1: Convert $0.35$ to a fraction in simplest form.

• Two decimal places $\implies$ denominator is $100$.
• Fraction: $\frac{35}{100}$.
• Both 35 and 100 divide by 5: $\frac{35 \div 5}{100 \div 5} = \frac{7}{20}$.

Worked Example 2: Convert $0.048$ to a fraction in simplest form.

• Three decimal places $\implies$ denominator is $1000$.
• Fraction: $\frac{48}{1000}$.
• Divide by $\text{HCF}(48, 1000) = 8$:
• $\frac{48 \div 8}{1000 \div 8} = \frac{6}{125}$.

Converting Fractions to Decimals

• Method A (Equivalent Base-10 Denominator): When the denominator can easily be multiplied to become $10, 100,$ or $1000$: $$\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 0.35$$ $$\frac{3}{25} = \frac{3 \times 4}{25 \times 4} = \frac{12}{100} = 0.12$$ $$\frac{11}{250} = \frac{11 \times 4}{250 \times 4} = \frac{44}{1000} = 0.044$$
• Method B (Short Division): When the denominator is not readily scalable, divide the numerator by the denominator using short division with decimal points: $$\frac{5}{8} = 5.000 \div 8 = 0.625$$

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