📐 Concept diagram

00-03 — Decimals

Phase: 0 — Arithmetic & Number Foundations Subject: 00-03 Prerequisites: 00-02 — Fractions Next subject: 00-04 — Percentages

Learning Objectives

By the end of this subject, you will be able to:

1. Identify the place value of each digit in a decimal number and read decimal numbers correctly
2. Distinguish between terminating and recurring decimals, and convert between fractions and decimals in both directions
3. Round decimal numbers to a specified number of decimal places or significant figures
4. Perform addition, subtraction, multiplication, and division with decimal numbers using place-value alignment
5. Express very large and very small numbers in scientific notation, and convert between standard and scientific forms

Core Content

1. What Is a Decimal?

A decimal is a way of writing fractions whose denominators are powers of 10 (10, 100, 1000, etc.) using the base-10 place-value system extended to the right of the units column.

The decimal point (a dot: .) separates the whole number part from the fractional part.

$   hundreds  tens  ones  .  tenths  hundredths  thousandths  ...
    100      10     1    .   1/10     1/100       1/1000      ...
$

Every position to the right of the decimal point is a fraction of a power of 10:

Example: The number 35.728 means:

$3 tens        = 3 × 10          = 30
5 ones        = 5 × 1           = 5
7 tenths      = 7 × (1/10)      = 0.7
2 hundredths  = 2 × (1/100)     = 0.02
8 thousandths = 8 × (1/1000)    = 0.008

Total: 30 + 5 + 0.7 + 0.02 + 0.008 = 35.728
$

How decimals connect to fractions: The number 35.728 can be written as the fraction 35,728/1000 (three decimal places → denominator 10³ = 1000). This gives us the fraction-decimal bridge from 00-02.

Reading decimals: 35.728 is read as "thirty-five point seven two eight" or "thirty-five and seven hundred twenty-eight thousandths."

2. Place Value and Zeros in Decimals

Trailing Zeros

Zeros at the end of a decimal (after the last non-zero digit) do NOT change the value:

$0.5 = 0.50 = 0.500 = 0.5000 = ...
$

Why? Because 0.5 = 5/10, 0.50 = 50/100, 0.500 = 500/1000 — all equivalent fractions (divide each by 5, 50, 500 respectively and you get 1/2).

But trailing zeros MATTER for precision. 0.5 could mean "roughly half" (1 significant figure), while 0.500 might mean "exactly half to the nearest thousandth" (3 significant figures). More on this in the rounding section.

Leading Zeros

Zeros between the decimal point and the first non-zero digit ARE significant for place value:

0.005  —  the zeros tell us these are thousandths, not tenths
0.05   —  the zero tells us these are hundredths, not tenths

Without leading zeros, 0.5, 0.05, and 0.005 would all look the same. The zeros establish magnitude.

Embedded Zeros

Zeros between non-zero digits after the decimal point are significant:

1.203 — the zero in the hundredths place means there are 0 hundredths

3. Converting Between Fractions and Decimals

Fraction → Decimal

As introduced in 00-02: divide the numerator by the denominator.

Terminating decimals: The division ends (remainder becomes 0).

$3/8  = 3 ÷ 8 = 0.375           ← terminates
1/4  = 1 ÷ 4 = 0.25            ← terminates
7/20 = 7 ÷ 20 = 0.35           ← terminates
$

When does a fraction produce a terminating decimal? A fraction in lowest terms produces a terminating decimal if and only if the denominator's only prime factors are 2 and/or 5.

Why? Because 10 = 2 × 5, and when the denominator is made up only of 2s and 5s, you can multiply numerator and denominator to get a power of 10.

$3/8 = 3/(2³): multiply by 5³ to get 3×125 / 8×125 = 375/1000 = 0.375
1/20 = 1/(2²×5): multiply by 5 to get 5/100 = 0.05
$

Recurring (Repeating) Decimals

If the denominator has any prime factor other than 2 or 5, the decimal repeats forever.

$1/3  = 1 ÷ 3  = 0.33333...  = 0.3̄   (the 3 repeats forever)
1/6  = 1 ÷ 6  = 0.16666...  = 0.16̄  (only the 6 repeats)
1/7  = 1 ÷ 7  = 0.142857142857... = 0.1̄4̄2̄8̄5̄7̄  (6-digit repeating block)
$

Notation: A bar or dot over the repeating digit(s): - 0.3̄ means 0.33333... - 0.16̄ means 0.16666... - 0.1̄4̄2̄8̄5̄7̄ means 0.142857142857...

Why do non-2/5 denominators produce recurring decimals? When you do long division, at each step you bring down a zero and divide. With a denominator like 3, the remainders cycle: 1÷3 remainder 1 → bring down 0 → 10÷3 = 3 remainder 1 → bring down 0 → 10÷3 = 3 remainder 1 → forever. Since there are only finitely many possible remainders (0 to b−1), eventually a remainder repeats, and the cycle begins.

Decimal → Fraction

For terminating decimals: 1. Count decimal places (n). 2. Write the decimal without the point as the numerator. 3. Denominator = 10ⁿ. 4. Simplify.

$0.375 → 3 decimal places → 375/1000 → (÷125) → 3/8
0.04  → 2 decimal places  → 4/100     → (÷4)   → 1/25
2.15  → 215/100            → (÷5)    → 43/20
$

For recurring decimals: Converting recurring decimals to fractions is covered in depth in algebra (Phase 2). For now, memorise common ones:

$0.333... = 0.3̄ = 1/3
0.666... = 0.6̄ = 2/3
0.111... = 0.1̄ = 1/9
0.999... = 0.9̄ = 1   (surprising but true — 0.9̄ equals exactly 1)
$

⚠️ THIS IS CRITICAL — The fact that 0.9̄ = 1 is one of the most counterintuitive facts in mathematics. Proof: let x = 0.999... Then 10x = 9.999... Subtract: 10x − x = 9.999... − 0.999... → 9x = 9 → x = 1. This is NOT an approximation — 0.9̄ and 1 are two ways to write the same number.

4. Rounding

Rounding replaces a number with a nearby, simpler number. We round to a specified number of decimal places (dp) or significant figures (sf).

Rounding to Decimal Places (dp)

"Round to n decimal places" means: look at the (n+1)-th decimal digit. If it's ≥ 5, round up; if < 5, round down (truncate).

Algorithm: 1. Identify the digit in the position you're rounding to. 2. Look at the digit immediately to its right. 3. If that digit is 5, 6, 7, 8, or 9: add 1 to the rounding digit (carry if needed). 4. If that digit is 0, 1, 2, 3, or 4: leave the rounding digit unchanged. 5. Drop all digits to the right.

Examples:

Round 3.14159 to 2 dp:
  Rounding digit = 4 (2nd decimal place)
  Next digit = 1 (3rd decimal place)
  1 < 5, so keep 4
  Result: 3.14

Round 2.71828 to 3 dp:
  Rounding digit = 8 (3rd decimal place)
  Next digit = 2 (4th decimal place)
  2 < 5, so keep 8
  Result: 2.718

Round 4.386 to 2 dp:
  Rounding digit = 8 (2nd decimal place)
  Next digit = 6 (3rd decimal place)
  6 ≥ 5, so round up: 8→9
  Result: 4.39

Round 9.997 to 2 dp:
  Rounding digit = 9, next digit = 7 ≥ 5, so round up
  9→10, so carry: the 9 in the tenths place becomes 10
  Wait — that means rounding 9.997 to 2dp gives us:
  Round at 9 (second decimal: hundredths), next is 7 → round up
  9.99(7) → 9.99 + 0.01 = 10.00
  Result: 10.00

The Tie-Breaking Convention (Rounding 5)

When the next digit is exactly 5 (with nothing after it), the convention is to round to the nearest even digit. This is called "banker's rounding" or "round half to even."

Round 2.5 to 0 dp → 2  (2 is the nearest even integer)
Round 3.5 to 0 dp → 4  (4 is the nearest even integer)
Round 1.25 to 1 dp → 1.2  (2 is even)
Round 1.35 to 1 dp → 1.4  (4 is even, since 1.35 rounds up and the result is 1.4)

Note: Some contexts use "round half up" (always round 5 up). Both are used in practice — specify which convention you're using.

Rounding to Significant Figures (sf)

Significant figures count all digits except leading zeros. Trailing zeros after the decimal point ARE significant. The rules:

1. All non-zero digits are significant.
2. Zeros BETWEEN non-zero digits are significant (e.g., 2003 has 4 sf).
3. Leading zeros (before the first non-zero digit) are NOT significant (e.g., 0.0034 has 2 sf).
4. Trailing zeros AFTER the decimal point ARE significant (e.g., 5.200 has 4 sf).
5. Trailing zeros in a whole number without a decimal point are ambiguous (e.g., 2400 — is it 2, 3, or 4 sf? Use scientific notation to clarify: 2.4 × 10³ has 2 sf, 2.400 × 10³ has 4 sf).

Algorithm for rounding to n sf: 1. Identify the n-th significant digit (count from left, skipping leading zeros). 2. Look at the next digit. 3. Apply the rounding rule (≥ 5 round up, < 5 round down). 4. If rounding up causes carries, propagate them. 5. Replace all digits after the n-th significant digit with zeros (or drop them if after the decimal point).

Examples:

Round 0.003827 to 2 sf:
  Sig digits: 3 (1st), 8 (2nd)
  Next digit: 2 (< 5, so keep 8)
  Result: 0.0038

Round 45,832 to 3 sf:
  Sig digits: 4 (1st), 5 (2nd), 8 (3rd)
  Next digit: 3 (< 5, so keep 832 as is, then zero out after)
  Result: 45,800

Round 0.04796 to 3 sf:
  Sig digits: 4 (1st), 7 (2nd), 9 (3rd)
  Next digit: 6 (≥ 5, so 9→10, carry)
  0.0479(6) → 0.0479 + 0.0001 = 0.0480
  Result: 0.0480

Common pitfall: Rounding 0.04796 to 3 sf as 0.047 is WRONG. You must also round the 9 up to 10, which carries into the 7.

5. Decimal Operations

Addition and Subtraction

Rule: Align the decimal points vertically. This ensures digits with the same place value line up in the same column. Fill empty positions with zeros as placeholders.

Example 1: Calculate 23.45 + 6.782

   2 3 . 4 5 0    ← fill with zero for alignment
+     6 . 7 8 2
---------------
   3 0 . 2 3 2

Step by step: - Thousandths: 0 + 2 = 2 - Hundredths: 5 + 8 = 13 → write 3, carry 1 - Tenths: 1 + 4 + 7 = 12 → write 2, carry 1 - Ones: 1 + 3 + 6 = 10 → write 0, carry 1 - Tens: 1 + 2 + 0 = 3

Answer: 30.232

Example 2: Calculate 15.3 − 8.74

   1 5 . 3 0    ← fill with zero
−     8 . 7 4
---------------
      6 . 5 6

Borrowing needed in tenths: 3 borrow from 5 ones → ones becomes 4, tenths becomes 13. 13 − 7 = 6 (tenths). 4 − 8 = can't, borrow from tens: 14 − 8 = 6 (ones).

Answer: 6.56

Multiplication

Rule: Multiply as if both numbers were whole numbers (ignore the decimal points), then place the decimal point in the answer so that the total number of decimal places equals the sum of the decimal places in both factors.

Why this works: 2.3 × 1.5 means (23/10) × (15/10) = (23 × 15)/(10 × 10) = (23 × 15)/100. Multiplying the whole numbers 23 and 15 gives the numerator, and dividing by 100 (because each factor was divided by 10) puts the decimal point back.

Example 1: Calculate 3.24 × 2.5

Step 1: Ignore decimals, multiply as whole numbers
      324
    ×  25
    -----
     1620   ← 324 × 5
     6480   ← 324 × 2, shifted
    -----
     8100

Step 2: Count decimal places
  3.24 has 2 decimal places
  2.5  has 1 decimal place
  Total: 3 decimal places

Step 3: Place the decimal point
  8100 → 8.100 = 8.1

Answer: 8.1

Common pitfall: The number of decimal places in the product is the SUM, not the product. 3.24 has 2 dp, 2.5 has 1 dp → 3 dp total, not 2 × 1 = 2 or anything else.

Example 2: Calculate 0.03 × 0.004

3 × 4 = 12

Decimal places: 0.03 (2 dp) + 0.004 (3 dp) = 5 dp
12 with 5 decimal places = 0.00012

Answer: 0.00012

Mental check: 0.03 is about 3 hundredths. 0.004 is 4 thousandths. Their product should be very small — 12 hundred-thousandths. 0.00012 ≈ 0.0001 ✓

Division

Case 1: Dividing by a whole number

Divide normally using long division, but bring the decimal point straight up into the quotient.

Example: Calculate 8.52 ÷ 4

$      2 . 1 3
    ---------
4 ) 8 . 5 2
    8
    -
    0 5
      4
      -
      1 2
      1 2
      --
       0
$

Answer: 2.13

Case 2: Dividing by a decimal

1. Move the decimal point in the divisor to the right until it becomes a whole number.
2. Move the decimal point in the dividend by the SAME number of places.
3. Bring the decimal point straight up into the quotient.
4. Divide as normal.

Example: Calculate 6.25 ÷ 0.5

$Step 1: Move decimal in divisor (0.5) one place right → 5
Step 2: Move decimal in dividend (6.25) one place right → 62.5
Step 3: Divide 62.5 ÷ 5

       1 2 . 5
     ---------
5 ) 6 2 . 5
    5
    -
    1 2
    1 0
    --
     2 5
     2 5
     --
      0
$

Answer: 12.5

Why does this work? Moving the decimal point multiplies both numbers by the same power of 10. Since division means (a × 10ⁿ) ÷ (b × 10ⁿ) = a ÷ b, the quotient is unchanged.

Check: 12.5 × 0.5 = 6.25 ✓

6. Scientific Notation (Standard Form)

Scientific notation expresses numbers as:

$a × 10ⁿ
$

where 1 ≤ |a| < 10 and n is an integer.

It's used to write very large and very small numbers compactly and to clearly show the number of significant figures.

Examples:

$3,200,000 = 3.2 × 10⁶     (move decimal 6 places left)
0.000045  = 4.5 × 10⁻⁵    (move decimal 5 places right)
299,792,458 = 2.99792458 × 10⁸   (speed of light in m/s)
0.0000000000000000001602 = 1.602 × 10⁻¹⁹   (electron charge in coulombs)
$

Converting to scientific notation:

For large numbers (≥ 10):
  Move decimal left until you have one digit before the point.
  Count how many places you moved — that's n (positive).

  45,300 → move 4 places → 4.53 × 10⁴

For small numbers (< 1):
  Move decimal right until you have one non-zero digit before the point.
  Count how many places you moved — that's |n| (n is negative).

  0.00072 → move 4 places → 7.2 × 10⁻⁴

Converting from scientific notation:

$3.8 × 10⁵  → move decimal 5 places right → 380,000
2.1 × 10⁻³ → move decimal 3 places left  → 0.0021
$

Operations in scientific notation:

Multiplication: (a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ

$(3 × 10⁴) × (2 × 10⁵) = (3×2) × 10⁴⁺⁵ = 6 × 10⁹
$

Division: (a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ

$(8 × 10⁷) ÷ (2 × 10³) = (8÷2) × 10⁷⁻³ = 4 × 10⁴
$

⚠️ THIS IS CRITICAL — Scientific notation is the standard way scientists and engineers communicate measurements. It is used in every later phase of this curriculum whenever we deal with very large numbers (astronomical distances, Avogadro's number) or very small numbers (atomic sizes, probabilities).

Key Terms

• Decimal operations
• Decimals
• Division by a decimal
• Rounding
• Scientific notation
• Terminating decimals

Worked Examples

Example 1: Fraction to Decimal with Long Division

Problem: Convert 7/12 to a decimal. Is it terminating or recurring?

Solution:

$       0 . 5 8 3 3 ...
     ----------------
12 ) 7 . 0 0 0 0 ...
      6 0
      --
      1 0 0
        9 6
        ---
          4 0
          3 6
          ---
            4 0
            3 6
            ---
              4 ...
$

The remainder 4 repeats, so the decimal repeats: 7/12 = 0.58333... = 0.583̄

Since 12 = 2² × 3 (contains a factor 3, which is not 2 or 5), this must be recurring.

Answer: 0.583̄ (recurring)

Example 2: Rounding with Carries

Problem: Round 0.03996 to 3 significant figures.

Solution:

Identify the significant digits. Leading zeros (0.0) are not significant.
1st significant digit: 3
2nd significant digit: 9
3rd significant digit: 9

Look at the next digit (the 4th significant digit): 6

Since 6 ≥ 5, round the 3rd significant digit (9) up by 1.
9 + 1 = 10 → write 0, carry 1 to the next digit (the 2nd sig digit, which is also 9)
9 + 1 = 10 → write 0, carry 1 to the next digit (the 1st sig digit, which is 3)
3 + 1 = 4

The result after the 3rd significant figure: 0.0400
(4 significant figures? No — the leading zero is not significant, so 0.0400 has 3 sf: 4, 0, 0.)

Answer: 0.0400 (3 sf)

Example 3: Decimal Multiplication with Cross-Check

Problem: Calculate 0.45 × 3.6

Solution:

$Method 1: Whole number multiplication, then place decimal
  45 × 36 = 45 × 30 + 45 × 6 = 1350 + 270 = 1620

  0.45 has 2 dp, 3.6 has 1 dp → total 3 dp
  1620 → 1.620 = 1.62

Method 2: Scientific notation / fraction check
  0.45 × 3.6 = (45/100) × (36/10) = (45×36)/(1000) = 1620/1000 = 1.62 ✓
$

Answer: 1.62

Example 4: Decimal Division by a Decimal

Problem: Calculate 0.084 ÷ 0.035

Solution:

$Step 1: Make divisor a whole number
  Move decimal in 0.035 three places right → 35
  Move decimal in 0.084 three places right → 84

Step 2: Divide 84 ÷ 35

        2 . 4
      -------
35 ) 8 4 . 0
      7 0
      ---
      1 4 0
      1 4 0
      -----
          0

Step 3: Simplify/check
  0.084 ÷ 0.035 = 2.4
  Check: 0.035 × 2.4 = (35×24)/100000... 
  35 × 2.4 = 35 × 2 + 35 × 0.4 = 70 + 14 = 84
  0.035 × 2.4 = 0.084 ✓
$

Answer: 2.4

Example 5: Scientific Notation

Problem: The distance from Earth to the Sun is approximately 149,600,000,000 metres. Express this in scientific notation with 3 significant figures.

Solution:

Step 1: Write the number with decimal point after the first non-zero digit
  1.49600000000

Step 2: Count places moved
  From 149,600,000,000 to 1.496... we moved 11 places left
  So the exponent is +11

Step 3: Apply 3 significant figures
  1.49600000000 × 10¹¹ → round to 3 sf
  1st sig: 1, 2nd: 4, 3rd: 9
  Next: 6 (≥ 5, so 9→10, carry)
  1.4(9+1 = 10 → 1.5)
  So: 1.50 × 10¹¹

Answer: 1.50 × 10¹¹ m

Practice Problems

(Answers are below. Try each problem before checking.)

Problem 1: Convert 11/16 to a decimal.

Problem 2: Convert 0.875 to a fraction in lowest terms.

Problem 3: Round 3.14159 to (a) 3 decimal places and (b) 4 significant figures.

Problem 4: Calculate: 12.38 + 5.724 − 3.6

Problem 5: Calculate: 0.072 × 0.35

Problem 6: Calculate: 9.75 ÷ 0.25

Problem 7: Express 0.0000673 in scientific notation.

Problem 8: Calculate (3.2 × 10⁴) × (5.0 × 10⁻²) and express in scientific notation.

Summary

1. Decimals extend the base-10 place-value system to the right of the decimal point, with positions representing tenths, hundredths, thousandths, etc.
2. Terminating decimals come from fractions whose denominators have only 2 and 5 as prime factors; all other fractions produce recurring decimals whose digit patterns repeat infinitely
3. Rounding to n decimal places looks at the (n+1)-th digit; rounding to n significant figures counts all non-zero digits and trailing zeros after the decimal point as significant, ignoring leading zeros
4. Decimal operations use the same algorithms as whole-number arithmetic, with careful attention to decimal point alignment (addition/subtraction) and decimal place counting (multiplication/division)
5. Division by a decimal is converted to division by a whole number by moving the decimal point in both divisor and dividend the same number of places
6. Scientific notation (a × 10ⁿ with 1 ≤ |a| < 10) compactly represents very large and very small numbers, and clearly communicates significant figures

Pitfalls

• Misaligning decimal points when adding or subtracting. The decimal points must be vertically aligned so that tenths are under tenths, hundredths under hundredths, etc. Writing 23.45 + 6.782 without alignment leads to nonsense.
• Miscounting decimal places in multiplication. The number of decimal places in the product is the sum of the decimal places in the factors, not the product. For 0.03 × 0.004: 2 dp + 3 dp = 5 dp, giving 0.00012.
• Rounding intermediate results. Rounding should be done only at the final step. Rounding 3.14159 to 2 dp as 3.14 and then multiplying gives a different (worse) answer than multiplying first and rounding last.
• Confusing significant figures with decimal places. 0.0047 has 2 significant figures (4 and 7) but 4 decimal places. The leading zeros are placeholders, not significant digits.
• Forgetting to move the decimal in both numbers when dividing by a decimal. When computing 6.25 ÷ 0.5, you must move the decimal in both the divisor and dividend the same number of places (→ 62.5 ÷ 5). Moving only one produces a wrong quotient.

Quiz

Answer each question, then read the explanation for your choice.

Q1: What is the place value of the digit 7 in the number 3.457?

A) tenths B) hundredths C) thousandths D) ones

Q2: Convert 5/6 to a decimal.

A) 0.83 B) 0.8333... C) 0.833 D) 0.8

Q3: Round 4.7962 to 3 significant figures.

A) 4.79 B) 4.80 C) 4.8 D) 4.796

Q4: Calculate: 0.6 × 0.08

A) 0.48 B) 0.048 C) 0.0048 D) 4.8

Q5: Calculate: 7.2 ÷ 0.09

A) 0.8 B) 8 C) 80 D) 800

Q6: Which of the following fractions will produce a terminating decimal?

A) 1/3 B) 5/6 C) 7/40 D) 2/21

Q7: Express 0.000308 in scientific notation.

A) 3.08 × 10⁻⁴ B) 3.08 × 10⁻³ C) 3.08 × 10⁻⁵ D) 3.08 × 10⁻⁶

Next Steps

Move on to 00-04 — Percentages to learn about converting between fractions, decimals, and percentages, and applying percentage calculations to real-world problems.

Q5: Which of the following is true?

A) 0.9̄ < 1 B) 0.9̄ = 1 C) 0.9̄ ≈ 1 (approximately, but not exactly) D) 0.9̄ is undefined

Answer: B) 0.9̄ = 1

Let x = 0.999... Then 10x = 9.999... Subtract: 10x − x = 9.999... − 0.999... → 9x = 9 → x = 1. 0.9̄ and 1 are the same number.