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Parallelogram 42 • Level 1 • 19 Jun 2025Why you can't divide by zero

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

As it's nearly summer, and the school year will soon be over, this is our last Level 1 Parallelogram of the year. However, that doesn’t mean that it’s the end of mathematics for the year!

If you are taking maths seriously, and if you are having fun with it (being serious and having fun can go together), then it is important that you keep the numbers and geometry parts of your brain at least a little bit busy over the summer.

I am hoping that your teacher will have set you some holiday work, but if you have missed any earlier Parallelograms then the summer is a great time to catch up.

1. Dividing by zero

There are lots of rules which help us understand mathematics, but maths is as much about breaking rules as it is about making rules.

There is, however, one rule in maths which must never be broken: don’t divide by zero!

But why?! Watch this video to find out:

(If you have problems watching the video, right click to open it in a new window)

1 mark

1.1 Which of these sums would result in the biggest answer?

  • 36÷12
  • 36÷2
  • 36÷12
  • 36÷14
  • (Not answered)
1 mark

1.2 Which of these statements is correct for positive numbers?

  • The smaller the divisor the bigger the answer
  • The smaller the divisor the smaller the answer
  • The bigger the divisor the bigger the answer
  • (Not answered)
1 mark

1.3 If I have a number and multiply it by 3, what must I do to return to the original number?

  • - 3
  • × 3
  • × 13
  • ÷ 13
  • (Not answered)

Multiplying by 13 is the same as dividing by 3.

So 3 and 13 are “multiplicative inverses”.

To undo × 3 you could ÷ 3 or you could × 13.

1 mark

1.4 What happens when you multiply a number by it's multiplicative inverse?

  • You can’t do this
  • The answer is always 1
  • The answer is always 0
  • The answer is always infinity
  • (Not answered)

Multiplying a number by it's inverse always results in 1, eg:

  • 2×12=1
  • 3×13=1
  • 4×14=1
  • 12345×112345=1
1 mark

1.5 What is the multiplicative inverse of zero?

  • 10
  • Infinity
  • 1
  • You can’t trick me, zero doesn’t have a multiplicative inverse
  • (Not answered)

A number multiplied by it's inverse always gives 1 and there is no way we can complete the calculation 0 × ? = 1.

So poor zero is left without a multiplicative inverse and thus you cannot divide by zero.

sad violins playing softly in the background for zero

2 marks

1.6 Which number is it's OWN multiplicative inverse?

  • -1
  • 1
  • 2
  • 10
  • Infinity
  • (Not answered)

A number multiplied by it's inverse always gives 1, and there is no number we can use to complete the calculation 0 × ? = 1.

So poor zero is left without a multiplicative inverse and thus you cannot divide by zero.

(sad violins playing softly in the background for zero)

2. A puzzle

Unlike dividing by zero, which is definitely impossible - this puzzle is possible, just a bit of a challenge!

2 marks

2.1 What is the missing number?

  • 6
  • 8
  • 9
  • 53
  • 54
  • (Not answered)
Show Hint (–1 mark)
–1 mark

Look at the top right number. How can the other three numbers in the square be used to create that top right number? Is there a pattern?

eg 9 ? - 5 = 67

How can you use the 8 to make this sum complete?

The pattern is multiply the bottom two numbers and subtract the top left to get the top right:

7 × ? - 4 = 52

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and collect your reward points.
  • You might earn a new badge...
  • Make sure you go through the solution sheet – it is massively important.
  • A score of less than 50% is ok – it means you can learn lots from your mistakes.
  • If you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.
  • This was our last Parallelogram of the year, but be sure to come back in September for more puzzles and problems.

Cheerio, Simon.