Real-world integers

Negative numbers are not abstract symbols on a page. They are how the world is measured every day , by weather scientists, bankers, divers, mountaineers, and even cricket scorers. This section is a tour of the places integers live.

Concept

Temperature

Most temperature scales , Celsius, Fahrenheit , measure temperature relative to a chosen freezing point. In Celsius, the freezing point of water is $0°$ C.

Above $0$ : positive (warm).
Below $0$ : negative (cold).

In India, temperatures from $-10°$ C (Drass, Ladakh) to $+50°$ C (Phalodi, Rajasthan) have been recorded.

Money and accounts

Banks use integers to record both money you have (credit, positive) and money you owe (debt, negative).

A balance of $+5000$ means $5000$ rupees in your account.
A balance of $-500$ means an overdraft , you owe the bank $500$ rupees.

When you deposit ₹ $1000$ into a balance of $-500$ : $-500 + 1000 = 500$ . Now you have ₹ $500$ .

Sea level

Earth's surface is measured relative to mean sea level.

Above sea level: positive altitude.
Below sea level: negative altitude.

The Dead Sea is at $-430$ m. Mount Everest's peak is at $+8848$ m. The total height difference between them is $8848 - (-430) = 9{,}278$ m.

Time

Although "time" usually flows forward, we can also use integers when we measure time relative to a reference event:

" $3$ hours before sunrise" $= -3$ on the time-axis where sunrise is $0$ .
" $2$ years ago" $= -2$ on a year-axis where this year is $0$ .
Years BCE and CE: $200$ BCE is like year $-200$ ; $200$ CE is $+200$ . (Note: there is no year $0$ in this convention.)

Game scores

In many quizzes and games, wrong answers cost points. So your running score can dip below zero:

Start at $0$ . Right answer: $+10$ . Wrong answer: $-3$ .
After a string of mistakes, your score might be $-9$ , you owe points.

Lift and floors

We've used this throughout: the ground floor is $0$ . Above is positive, below is negative.

Other useful settings

Stock market changes. A stock rises or falls by some rupees each day. Net change over a week is the sum of daily changes (signed integers).
Sports. Goal difference (goals scored minus goals conceded) is a signed integer used in football tables.
Direction. East/right as positive, west/left as negative , for car positions, ships, planes.

What integers cannot easily express

"How many apples are on the plate" , always non-negative.
"How long is this rope" , always non-negative.
"How many friends do I have" , non-negative.

The rule: if a quantity has a natural meaning of "less than nothing" or "below a reference", integers (with negatives) help. Otherwise stick with non-negative numbers.

Worked examples

Example 1. On a winter day in Shimla, the temperature was $-3°$ C in the morning and rose by $8°$ by afternoon. Find the afternoon temperature.

$-3 + 8 = +5$ . So $5°$ C.

Example 2. Asha's bank balance was $-200$ (overdraft). She deposited $\rupee 500$ . What is her balance now?

$-200 + 500 = 300$ . ₹ $300$ in credit.

Example 3. A submarine at $-150$ m descends another $80$ m. What is its new depth?

$-150 + (-80) = -230$ . It is $230$ m below sea level.

Example 4. Year $300$ BCE is what integer on the year-axis (CE positive)?

$-300$ .

Example 5. A quiz: $+5$ for correct, $-2$ for wrong. After $6$ correct and $4$ wrong, what is the score?

$6 \times 5 + 4 \times (-2) = 30 - 8 = 22$ .

Example 6. A football team's goal difference is computed as (goals for) $-$ (goals against). Scored $14$ , conceded $20$ . Goal difference?

$14 - 20 = -6$ .

Try it yourself

The temperature in Leh at midnight is $-12°$ C. By morning it has risen $7°$ . Morning temperature?
Surya's bank balance is $-150$ . He deposits $\rupee 400$ . New balance?
A diver is at $-25$ m and dives down $15$ m more. New position?
A mountain peak is at $+5000$ m. A nearby lake is at $-100$ m. Height difference?
Anushka's quiz: $5$ correct ( $+4$ each) and $3$ wrong ( $-1$ each). Score?
A stock changes by $-3, +5, -2, -1, +6$ over five days. Net change?
Build a number line from $-1000$ m to $+5000$ m. Mark: Indian Ocean floor ( $-3000$ m), sea level ( $0$ ), Mumbai ( $+8$ m), Bengaluru ( $+920$ m), Shimla ( $+2200$ m).
Word problem: Two stations are at altitudes $+250$ m and $-50$ m. A train takes $30$ minutes to travel between them. What is the difference in altitude?
Challenge: Astronomers sometimes label the year $1$ BCE as $0$ , with $1$ CE as $1$ . Under this convention, how many years are there from $5$ BCE to $5$ CE?

Activity

Family bank. With your family, design a fictional bank account. Start with a balance of $\rupee 1000$ . Over $5$ days, simulate deposits and withdrawals (with one day going overdrawn , balance below $0$ ). Record each day's balance as a signed integer. Plot the $5$ values on a number line and discuss: what is the lowest balance? What is the highest? What is the difference?