Lesson 1 of 6
Numbers, direction, and order
A number tells us both how much and, when it has a sign, which side of zero it lives on.
Core lesson · about 6 minutes · 3 guided parts
Imagine the school corridor as a number line. Your classroom is 0, rooms to the right are positive, and rooms to the left are negative. Walking three doors left means −3 even though you walked a distance of 3.
Your learning route
Build the idea one piece at a time
- The number line and absolute value
- Operations and precedence
- Follow a signed quantity through a real system
The number line and absolute value
Position has direction; distance does not.
Integers are whole numbers and their negatives. Moving right makes a number larger; moving left makes it smaller.
The absolute value |x| is the distance from x to zero, so it is never negative. Opposites such as 7 and −7 have the same absolute value.
Comparing signed numbers becomes easy on a drawn line: the point farther right is always larger.
The minus sign can describe a negative position or the operation of subtraction. In −5 it is part of the number, while in 8−5 it tells us to find a difference.
The distance between two number-line points a and b is |a−b|. For example, the distance from −4 to 3 is |−4−3|=7, which agrees with counting seven unit steps.
Zero is neither positive nor negative, but it is the reference that separates them. Moving from −2 to 0 is an increase even though the final value is not positive.
|x| = distance from x to 0Worked example
A lift starts at floor 2, goes down 5 floors, then up 1. Where is it, and how far is that floor from 0?
- Start at 2.
- Going down 5 gives 2 − 5 = −3.
- Going up 1 gives −3 + 1 = −2.
- The distance from 0 is |−2| = 2.
Answer: Floor −2; distance 2 floors from 0.
Operations and precedence
A long calculation needs traffic rules so everyone gets the same result.
Grouping symbols come first, then powers, then multiplication or division from left to right, then addition or subtraction from left to right.
Subtraction can be viewed as adding an opposite: a − b = a + (−b). This view prevents many sign errors.
Estimation is a quick safety check. If 49 × 21 is near 50 × 20 = 1000, an answer of 102.9 cannot be right.
Nested grouping is handled from the innermost group outward. In 2[3+(5−1)], calculate 5−1 first, then add 3, and only then multiply by 2.
The sign rules for multiplication come from repeated direction changes: a positive times a negative is negative, while multiplying by another negative reverses direction again and gives a positive.
An estimate can be a range instead of one rounded guess. Since 19 lies between 18 and 20, 19×7 must lie between 126 and 140, so 133 is believable and 113 is not.
grouping → powers → × or ÷ → + or −Worked example
Evaluate 18 − 2(3 + 4) + 2³.
- Work inside parentheses: 3 + 4 = 7.
- Evaluate the power: 2³ = 8.
- Multiply: 2 × 7 = 14.
- Finish left to right: 18 − 14 + 8 = 12.
Answer: 12
Practical deep dive for this lesson
Follow a signed quantity through a real system
A sign is useful because it records which side of a chosen reference a quantity lies on; the arithmetic then records each change without losing direction.
Start by choosing what zero means. For temperature, zero may be 0 °C; for a bank account, it may mean no money owed or owned; for a machine-learning score, it may mean no change from a baseline.
A positive number means one direction from that reference and a negative number means the opposite direction. The words above/below, gain/loss, forward/backward, and credit/debt are clues about the sign.
Addition combines a current position with a signed change. Adding +4 means move four units in the positive direction, while adding −4 means move four units in the negative direction.
Subtraction asks for a difference or removes a signed quantity. Rewriting subtraction as addition of the opposite makes the direction visible: 6−(−3)=6+3.
Absolute value answers a different question from the signed number. A temperature change of −7 °C has direction downward, while |−7|=7 °C reports only the size of the change.
Ordering is always about position on the number line. Among negative numbers, the one closer to zero is larger: −2 is larger than −9 even though 9 has the larger magnitude.
This is used in software that tracks balances, elevation, temperature anomalies, profit and loss, and model errors. The sign tells the program which way to update; the magnitude tells it how large the update is.
A final estimate catches mistakes. If a value near 10 falls by about 15, the answer must be a little below −5, so a large positive result cannot be reasonable.
new value = old value + signed change; distance moved = |change|Worked example
A freezer is at −6 °C. Its temperature rises 9 °C during loading, then the cooling system lowers it by 4 °C. What is the final temperature, and what total distance did the temperature travel?
- Use −6 as the starting position on a temperature number line.
- A rise of 9 is a positive change, so calculate −6+9=3.
- The freezer is temporarily at 3 °C; crossing zero does not stop the calculation.
- A lowering of 4 is a change of −4, so calculate 3+(−4)=−1.
- The final signed temperature is therefore −1 °C.
- Distance travelled ignores direction: the first movement has size |9|=9 and the second has size |−4|=4.
- Add those movement sizes: 9+4=13 °C of total travel.
- Notice that final displacement is only −1−(−6)=5 °C, which is different from total distance travelled.
Answer: Final temperature −1 °C; total temperature movement 13 °C.
Check your picture
Which is farther from zero, −9 or 6, and which number is larger?
Reveal answer
−9 is farther from zero; 6 is larger.
This lesson versus the whole chapter
Your topic-specific practical example is above.
The final lesson collects five longer chapter-wide workflows for machine learning, LLMs, trading, physics, and everyday decisions.
Formula shelf
Symbols with meaning
|x| = distance(x, 0)Distance from zero ignores direction.
a/b + c/d = (ad + bc)/bdUse equal-sized parts before adding.
(new − old)/old × 100%, old≠0Measures change relative to a nonzero starting value.
aᵐaⁿ = aᵐ⁺ⁿRepeated factors combine by adding their counts.
logb x = y ⇔ bʸ = xA logarithm asks for an exponent.
|A ∪ B| = |A| + |B| − |A ∩ B|Subtract the overlap counted twice.
|estimate − true|/|true|, true≠0Scales an error by a nonzero true quantity.
a=qm+r, 0≤r<mSeparates an integer into full groups and one standard remainder.
a≡b (mod m) ⇔ m | (a−b)Says two integers occupy the same place in an m-step cycle.
Before moving on
You are ready when you can…
- Compare signed numbers and calculate with correct operation order.
- Move confidently among fractions, ratios, decimals, rates, and percentages.
- Use powers, roots, logarithms, and scientific notation to reason about scale.
- Read set and logic notation, and distinguish a claim from its converse.
- Check answers with units, estimates, counterexamples, and simple proof patterns.
- Compute standard remainders and use congruences to reason about repeating cycles.