The arrival of zero and negative numbers

It seems obvious that we should have a symbol for "nothing". But for thousands of years humans did not. They had marks for $1, 2, 3, \dots$, names for "many", and ways to say a column was empty (a blank, a dot, a gap). The full symbol $0$, treated as a number you can compute with, was a much later invention. So were negative numbers. Once we had both, mathematics changed forever.

Concept

Zero plays two distinct roles.

1. Placeholder. In $204$, the $0$ tells us "no tens here". Without it, $204$ would collapse to $24$. This use of $0$ as a placeholder dates back to Babylonian and Mayan astronomy.

2. A number on its own. $0$ is what you get when you take any number and subtract itself: $5 - 5 = 0$. The Indian mathematician Brahmagupta (7th century CE) was the first to give clear rules for $0$ as a number:

   • $a + 0 = a$
   • $a - 0 = a$
   • $a \times 0 = 0$
   • $0 \div a = 0$ (if $a \ne 0$)
   • and crucially, $a \div 0$ is undefined.

The reason for that last rule: if $a \div 0 = b$, then by reversing, $b \times 0 = a$. But $b \times 0$ is always $0$, so this only works if $a = 0$ , and even then there is no unique answer for $b$. So dividing by zero is excluded from arithmetic.

Negative numbers appeared as soon as people tried to subtract a bigger number from a smaller one , say $3 - 7$. For centuries Greek and European mathematicians simply called such problems "impossible" or "absurd". Indian and Chinese mathematicians, dealing with debts and losses, were more open. Brahmagupta gave rules like:

• A debt minus zero is a debt.
• A fortune minus zero is a fortune.
• The product of two debts is a fortune. (i.e., $(-a)\times(-b) = ab$.)

Picture a number line with $0$ in the middle. Positive numbers stretch right; negative numbers stretch left. Every integer has a "mirror image" (its opposite or additive inverse): the mirror of $3$ is $-3$, and $3 + (-3) = 0$.

Negative numbers turn many problems into something solvable:

• A temperature of $4^\circ C$ falling by $10$ degrees becomes $-6^\circ C$, not "impossible".
• An account balance going $\mathbb{?}\,200$ overdrawn is $-200$.
• A point $50$ m below sea level has elevation $-50$ m.

Together, the integers $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$ form a complete arithmetic world where every subtraction is possible.

Sign rules for multiplication and division:

So $(-6) \times (-4) = 24$ and $(-15) \div 3 = -5$.

Worked examples

Example 1. Simplify $(-12) + 7 - (-5)$.

• $(-12) + 7 = -5$.
• $-5 - (-5) = -5 + 5 = 0$.

Example 2. Find $(-3) \times (-4) \times 5$.

• $(-3) \times (-4) = 12$.
• $12 \times 5 = 60$.

Example 3. A submarine at $-180$ m descends another $45$ m. Find its new depth.

• $-180 + (-45) = -225$ m.

Example 4. Brahmagupta's rule says debt $\times$ debt $=$ fortune. Verify with a small example: $(-2) \times (-3)$.

• $(-2) \times (-3)$. Two negatives multiply to positive, so $= 6$. ✓

Try it yourself

1. Plot $-4$, $-1$, $0$, $3$, $5$ on a number line.
2. Compute $-8 + 13$ and $13 - (-8)$.
3. Compute $(-7) \times (-6)$ and $(-7) \times 6$.
4. Compute $(-48) \div 6$ and $(-48) \div (-6)$.
5. A lift goes from floor $-3$ (basement $3$) to floor $7$. How many floors does it climb?
6. Use the additive inverse to solve $x + 9 = 2$.
7. Without computing, decide the sign of $(-2) \times (-3) \times (-4) \times (-5)$.
8. Why is $\dfrac{0}{0}$ not defined? (Hint: any number could "work" if it were.)

Activity / Insight

Travel through history. Look up when zero appeared in five different cultures (Babylonian, Mayan, Indian, Arab, European). Make a small timeline. You will find that India's contribution stands out: it was here that $0$ first became a full-fledged number with its own arithmetic, thanks to Brahmagupta around 628 CE. Without that step, computers, banking, and almost all modern science would be far harder to build.