Math Detours · Grade 7 · Around the corner · 21

Why you cannot divide by zero

12 ÷ 0 is not infinity and not zero. Ask what multiplication would have to do, and no answer survives.

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Tonight's idea

Division undoes multiplication. \(12 \div 3 = 4\) because \(4 \times 3 = 12\). Every division is a question in disguise: what number times 3 makes 12?

Now try \(12 \div 0\). The question becomes: what number times 0 makes 12? But any number times 0 is 0, never 12. No number answers the question. Zero does not work either, since \(0 \times 0\) is 0, not 12.

What about \(0 \div 0\)? The question is: what number times 0 makes 0? Now every number works. 5 works, and so does 73. A question with every possible answer is no more use than a question with none.

\(a \div b\) means the one number that, multiplied by \(b\), gives \(a\). When \(b = 0\) there is no such number (if \(a\) is not 0) or there are too many (if \(a\) is 0). So dividing by zero is undefined: it has no meaning, because no meaning works.

Watch one: creep up on zero

Perhaps the answer is "infinity". Divide 12 by smaller and smaller numbers and watch.

Divide 12 by10.10.010.001or by−1−0.1−0.01−0.001
Answer121201,20012,000−12−120−1,200−12,000
−3 −2 −1 1 2 3 12 ÷ 1 = 12 12 ÷ 0.5 = 24 12 ÷ 0.25 = 48 12 ÷ (−0.25) = −48 12 ÷ (−1) = −12 no point at 0
12 ÷ a number, for numbers near 0. The curve has two halves that fly apart, and no point at 0.

From the positive side the answers grow without limit. From the negative side they fall without limit. The two sides head in opposite directions, so they cannot agree on a value at 0. And infinity is not a number. It is a way of saying "this never stops growing".

Your turn

  1. Turn each division into a "what number times …" question, then answer it. (a) \(20 \div 5\) (b) \(0 \div 7\) (c) \(7 \div 0\) (d) \(0 \div 0\)
  2. (a) Work out \(6 \div 0.5\), \(6 \div 0.01\), \(6 \div (-0.01)\) and \(6 \div (-0.5)\). (b) Without a calculator, what is \(6 \div 0.000001\)?
  3. Each of these has no value for exactly one value of \(x\). Find it. (a) \(5 \div (x - 3)\) (b) \(8 \div (x + 4)\) (c) \(1 \div (2x - 10)\)
  4. Here is a "proof" that 1 = 2. Line 1: \(1 \times 0 = 2 \times 0\). Line 2: divide both sides by 0. Line 3: \(1 = 2\). Which line is wrong? Explain what goes wrong, using the idea that division undoes multiplication.

Stretch: invent a number

Suppose we invent a new number \(N\) and declare \(12 \div 0 = N\). Then \(N \times 0 = 12\). But \(0 + 0 = 0\), so \(N \times 0 = N \times (0 + 0)\). Expand the bracket. What impossible statement do you reach?

Answers

  1. (a) What number times 5 makes 20? 4. (b) What number times 7 makes 0? 0, and only 0. So \(0 \div 7 = 0\) is fine. (c) What number times 0 makes 7? None: undefined. (d) What number times 0 makes 0? Every number: undefined.
  2. (a) 12, 600, −600, −12. (b) 6,000,000. Dividing by a millionth is the same as multiplying by a million.
  3. The divisor must be 0. (a) \(x = 3\). (b) \(x = -4\). (c) \(2x = 10\), so \(x = 5\).
  4. Line 2. Line 1 is true: both sides are 0. Dividing by 0 would mean working back to the number that was multiplied by 0. But multiplying by 0 turns every number into 0, so there is no way back: 0 could have come from 1, from 2, or from anything. This is the \(0 \div 0\) case, where every answer fits.

Stretch. \(N \times (0 + 0) = N \times 0 + N \times 0 = 12 + 12 = 24\). But it also equals \(N \times 0 = 12\). So \(12 = 24\). Inventing an answer for \(12 \div 0\) breaks ordinary arithmetic, which is why nobody does it.

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