Integers: frost and debt · Lập trình với Martin

Numbers below zero

“It is minus five outside.” “I am in the minus, I owe you twenty.” “The lift goes to minus one.” Negative numbers you know from ordinary speech — now they get a place on the number line.

Until now your numbers started at zero and grew to the right: 0, 1, 2, 3… But what is LESS than nothing? Temperature below miễn phízing. Money you do not have, but you owe. A floor below ground level.

Negative numbers we write with a minus sign: −5, −20, −1. They sit on the number line to the left of zero — mirrored to the positive ones. Together with zero and the positive numbers they make integers: …, −3, −2, −1, 0, 1, 2, 3, …

Two things you need to get into your hands today:

Comparing: on the line the bigger number is the one further to the right. That is why −2 > −7. Frost minus two is “warmer” than minus seven!

Opposite numbers: 5 and −5 sit equally far from zero, each on a different side. Their sum is zero — twenty of earnings wipes twenty of debt.

Ema and Jonáš read the forecast: “At night −7 °C, in the day −2 °C.” Jonáš: “So in the day it will be a bigger miễn phíze, two is less than seven… no, wait.” Ema draws a thermometer in the snow — a vertical line with zero in the middle: “Minus seven is DEEPER below zero. The further from zero down, the bigger the frost.” Jonáš writes both numbers on the line: −7 is lower, −2 higher, closer to zero. “So in the day it warms by five degrees, even though it will still be frost.” Exactly — with negative numbers the one closer to zero wins.

💡

When you are stuck with negative numbers, draw a line. A second of drawing saves a minute of mess — on the line the bigger one is always the one further to the right (or higher on a thermometer).

The line, comparing and opposite numbers

The number line of integers:

−5 —— −4 —— −3 —— −2 —— −1 —— 0 —— 1 —— 2 —— 3 —— 4 —— 5

Zero is the middle. Positive numbers to the right, negative to the left. A thermometer is the same line, just standing up: plus at the top, minus at the bottom.

Comparing — there is only one rule: bigger is what is further RIGHT on the line.

Positive > negative, always: 1 > −100. (Even one koruna in your pocket is more than a hundred of debt.)
Zero > every negative: 0 > −3.
Between negatives the one closer to zero wins: −2 > −7. Watch, with negatives it flips against habit: seven is a “bigger digit”, but −7 is a smaller number — deeper in frost, deeper in debt.

An opposite number you get by flipping over zero: to 5 the opposite is −5, to −12 the opposite is 12. The opposite number of zero is zero itself.

The sum of opposite numbers is 0. You earn 50, you spend 50 → you have zero. This fact is the base of the next lesson, where negative numbers start being added and subtracted.

Where you meet negative numbers: temperature (−15 °C), a lift (−2 = the second underground), an account (−200 Kč = debt), the sea (−400 m = depth), a score in games.

Compare −3 and −8

I draw a bit of the line: −8 … −3 … 0.

The number −3 is closer to zero, so further to the right → −3 > −8.

Through a thermometer: minus three is a weaker frost than minus eight — three degrees below zero is “warmer” than eight below zero.

Through money: owing 3 korunas is better (bigger wealth!) than owing 8. A smaller debt = a bigger number. With negative numbers everything is measured by distance from zero.

Order: 2, −5, 0, −1, 4

I place them on the line (or I picture it):

−5 is furthest left, then −1, then 0, then 2, then 4.

From smallest: −5 < −1 < 0 < 2 < 4.

A check with the rules: negatives before zero (both are), between negatives the smaller is the one further from zero (−5 before −1), then zero, then positives by size. The line orders it for you.

By how much did it warm up?

In the morning it was −6 °C, in the afternoon +3 °C. By how many degrees did it warm up?

Along the line from −6 to +3: from −6 to zero is 6 steps, from zero to 3 are 3 steps.

Together 6 + 3 = 9 degrees.

Watch the trap: 6 − 3 = 3 is wrong — the temperature crossed THROUGH zero, so both distances add. Steps along the line will not betray you; next time we will learn to count the same without drawing.

⚠️

With negative numbers “a bigger digit = a bigger number” does not hold. −7 is SMALLER than −2, even though 7 > 2. The minus sign mirrors everything: the further left from zero, the smaller the value. Always compare place on the line, not digits.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Bắt đầu with ten. When it goes well, add more.

Try the first counting with negative numbers — help yourself with a line or a thermometer. Write a negative result with a minus, for example -4.

Practise the questions

On paper

Draw a proper line across the whole width of the paper — it will serve the next lesson too.

Draw a number line from −10 to 10 (by ones, with zero in the middle) and mark on it: −7, −3, 0, 4, 8.
Compare (write < or >): −4 and 2; −9 and −1; 0 and −6; −5 and −50.
Write the opposite numbers of: 7, −12, 0, 25. For each pair check that the sum makes zero.
Temperatures in cities: Prague −2 °C, Ostrava −8 °C, Brno 1 °C, Pilsen −5 °C. Order the cities from the coldest and count along the line by how many degrees it is warmer in Brno than in Ostrava.

Now you 💪

I can place a negative number on the line and I know that bigger is always the one further to the right.
I can compare two negative numbers by closeness to zero: −2 > −7.
I can find the opposite number and I know that it adds with the original to zero.

✅

Done when: You read, write and compare integers including negatives, you draw them on a line and you count a temperature difference by stepping through zero.

What to take from this lesson

Negative numbers sit to the left of zero: frost, debt, an underground floor, depth.
Integers: …, −2, −1, 0, 1, 2, … Zero is neither positive nor negative.
Bigger is the number further to the right on the line: 1 > −100, −2 > −7.
Opposite numbers (5 and −5) are a mirror through zero and add to zero.
A temperature difference through zero = steps to zero + steps from zero (from −6 to +3 is 9 degrees).