Decimal Places & Significant Figures

What error occurred in Spain’s S-80 submarine programme?

A decimal place error caused the submarine to be 70 tonnes overweight.

Why was this dangerous?

The submarine risked never resurfacing after diving.

What was the cost of the programme?

€2.4 billion for four submarines.

What did Spain do to fix the mistake?

Signed a €12 million deal with Electric Boat (USA) to help reduce the weight.

What is the essential rule when adding or subtracting decimals?

Line up the decimal points.

Why fill in “missing zeros”?

To make columns align correctly for addition/subtraction.

Example shown: 35.2 + 4.252 + 13.04 + 3.891 + 0.098

Correctly aligned and summed to 26.336.

What is the rule for multiplying decimals?

Ignore decimals during multiplication, then add the number of decimal places from each number to determine where the decimal goes.

Example: 8.23 × 3.2

• 8.23 has 2 decimal places

• 3.2 has 1 decimal place

• Answer has 3 decimal places.

What is the intermediate multiplication carried out as?

823 × 32 = 26,336 (then place decimal → 26.336).

Typical practice problems?

a) 0.002 × 1.25
b) 0.2 × 12.5
c) 0.02 × 0.000125

What is the key restriction when dividing decimals?

You may only divide by a whole number.

How to turn the divisor into a whole number?

“Bounce” the decimal to the right until no decimal remains.

What must you do to the dividend?

Bounce its decimal the same number of places.

Example: 0.12 ÷ 0.004

Bounce 3 places → 120 ÷ 4 = 30.

Practice problems given?

a) 0.0144 ÷ 0.09
b) 1.44 ÷ 0.0009
c) 0.144 ÷ 0.009

What is the general rounding rule?

If the next digit is ≥ 5, round up; otherwise leave as is.

Example: Round 16,498 to the nearest 10.

16,500.

Round 16,498 to nearest 1,000.

16,000.

Round 16,498 to nearest 10,000.

20,000.

Round 0.0018089 to 5 decimal places.

0.00181.

Round 0.0018089 to 3 decimal places.

0.002.

Round 0.0018089 to 1 decimal place.

0.0.

What is the method for rounding to decimal places?

• Start at decimal point.

• Count out the required number of places.

• Round based on the next digit (≥5 rule).

Example: 3.0585

– To 1 DP → 3.1
– To 2 DP → 3.06
– To 3 DP → 3.059

Example: 0.0498

– To 1 DP → 0.0
– To 2 DP → 0.05
– To 3 DP → 0.050

Practice problems: Convert values to 2 decimal places:

• 45.8065
• 0.04908
• 6743.087
• 0.527231
• 0.000319504

What is crucial difference between decimal places and significant figures?

• Decimal places start counting at the decimal point
• Significant figures start at the first non-zero digit

Rule for rounding sig figs?

Same as decimal places—round the next digit if ≥5.

Why must you maintain the order of magnitude?

To ensure the overall size of the number remains correct.

Example: 3.0585

– To 1 s.f. → 3
– To 2 s.f. → 3.1
– To 3 s.f. → 3.06

Example: 0.0498

– To 1 s.f. → 0.05
– To 2 s.f. → 0.050

Practice problems (convert to 2 significant figures):

• 45.8065
• 0.04908
• 6743.087
• 0.527231
• 0.000319504

When rounding to decimal places, where do you start counting?

At the decimal point.

When rounding to significant figures, where do you start counting?

At the first non-zero digit.

Why do significant figures matter in science?

They preserve correct precision and magnitude of measured data.