Below zero , meeting negative numbers

Why would anyone need a number less than zero? After all, you can't have $-3$ chapatis on a plate. But the moment we start measuring quantities that can go down below a reference level, negative numbers become essential.

Concept

Think of the lift in a tall building. The ground floor is " $0$ ". The first floor is " $+1$ ", the second " $+2$ ", and so on going up. If the building also has a basement, the floor below ground is " $-1$ " (one below ground), the next is " $-2$ ", and so on going down.

This same idea , a starting point with a "positive" side and a "negative" side , turns up everywhere:

Temperature. The freezing point of water is $0°$ C. Hotter than that is positive. Colder is negative. Delhi might hit $-1°$ C on a cold winter night.
Sea level. The surface of the sea is $0$ . Above sea level is positive. Below sea level is negative. The Dead Sea is at about $-430$ m.
Money. Your balance can be positive (you have money) or negative (you owe money , a debt).
Score. Some games allow negative scores. Losing $5$ points from $0$ leaves you at $-5$ .
Lift floors below ground in a multi-storey car park: $-1$ , $-2$ , $-3$ .

In every case there is a natural "zero" , a reference point , and quantities on one side are called positive, quantities on the other side are called negative.

Writing negative numbers

A negative number is written with a small minus sign in front: $-1$ , $-7$ , $-42$ . Read as "negative one" or "minus seven".

A positive number can be written either with no sign or with a plus sign: $5$ or $+5$ . Both mean the same.

Zero is neither positive nor negative. It is its own thing , the dividing line.

Opposites

Every positive number has an opposite (also called its negative): the number with the same size but on the other side of zero. The opposite of $7$ is $-7$ . The opposite of $-3$ is $+3$ .

Visually, opposites are the same distance from zero, but in opposite directions.

A subtle but important fact

The negative number $-100$ is smaller than $-1$ . Why? Because $-100$ is further "down" , further to the left of zero. Think of bank balances: owing ₹ $100$ (i.e., balance $-100$ ) is worse than owing ₹ $1$ (balance $-1$ ). So $-100 < -1$ .

In general, for negative numbers, the one with the bigger "size" (bigger absolute value) is smaller. Always picture the number line.

Worked examples

Example 1. Bela parks her car on the $3$ rd floor underground. Write the floor as an integer.

$-3$ .

Example 2. A lake's surface is at altitude $0$ . The lake bottom is $40$ m below. Write the bottom's altitude.

$-40$ m.

Example 3. Yesterday the temperature was $-2°$ C. Today it is $5°$ C. Which is warmer?

$5°$ C is warmer. $5 > -2$ .

Example 4. Order from smallest to largest: $-7, 3, 0, -1, 8, -10$ .

$-10 < -7 < -1 < 0 < 3 < 8$ .

Example 5. What is the opposite of $-12$ ?

$+12$ (or just $12$ ).

Example 6. A submarine is at $-200$ m. It rises to $-50$ m. Did it go up or down? By how much?

It went up by $200 - 50 = 150$ m. The depth went from $200$ m below to $50$ m below.

Try it yourself

Bela enters the lift on floor $-2$ and goes up by $5$ floors. What floor is she on?
The temperature at $6$ a.m. is $-3°$ C. By noon it is $8°$ C. By how much did it rise?
Order: $-15, 0, 5, -1, -20, 10$ from smallest to largest.
The Mariana Trench is about $11{,}000$ m below sea level. Write its altitude.
What is the opposite of $0$ ?
Which is larger: $-50$ or $-5$ ? Why?
Saurabh has a debt of ₹ $200$ . Write his balance as an integer.
Investigate: a stock price changes by $+5, -3, +2, -7, -1$ over five days. What is the net change?

Activity

Lift map. Sketch a tall building with floors from $-5$ (deep basement) to $+10$ . Label the ground floor " $0$ ". Mark these on the lift: "school office" at $-2$ , "swimming pool" at $-4$ , "library" at $+3$ , "rooftop garden" at $+10$ . Calculate: from school office to rooftop garden, how many floors up? From swimming pool to library, how many floors up?