Integers — operations, number line, absolute value

What Are Integers?

Why integers matter: Natural numbers $\{1,2,3,\ldots\}$ can't handle subtraction like $3- 5$ . Integers complete the system so every subtraction has answer.

The Number Line

Key features:

Origin: 0 is the center
Direction: Right = positive, Left = negative
Distance: Steps between points
Ordering: $a < b$ means " $a$ is left of $b$ "

Operations on Integers

Addition

Rule from movement:

Adding a positive number → move right
Adding a negative number → move left

Why this step? Each "move left by 5" is exactly what adding $-5$ means. The number line makes the rule visual.

Derivation of sign rules:

Same signs: If both positive or both negative, add magnitudes and keep the sign.
Why? Moving in the same direction accumulates distance.
$(-a) + (-b) = -(a + b) \quad \text{both move left}$
Opposite signs: Subtract the smaller magnitude from the larger, take the sign of the larger. If the magnitudes are equal, the result is 0.
Why? One movement cancels part of the other. The "winner" determines the final side; a tie lands you exactly at 0.
$a + (-b) = \begin{cases} a - b & \text{if } a > b \\ 0 & \text{if } a = b \\ -(b - a) & \text{if } b > a \end{cases}$

Subtraction

Rewrite as addition: $a - b = a + (-b)$

Why? Subtracting is the same as adding the opposite. "Take away 5" = "add negative 5" (move left 5).

Why this step? Converting to addition lets us use the single addition rule. "Subtracting a negative" becomes "adding a positive" because $-(-6) = 6$ .

Multiplication

Rule from repeated addition: $a \times b = \underbrace{a + a + \cdots + a}_{b \text{ times}} \quad \text{(if } b > 0\text{)}$

Sign rules:

(+) × (+) = (+): Adding positives stays positive
(+) × (−) = (−): Adding negatives (opposite direction) gives negative
(−) × (+) = (−): Same as above (commutative)
(−) × (−) = (+): WHY? Think of "taking away debt". If you remove 3 debts of $5 each, you gain $15.

Formal derivation of $(-) \times (-) = (+)$ :

Start with the additive inverse property: $a + (-a) = 0$ for any $a$ .

Let $a = 3$ and $b = 5$ : $3 \times 5 + 3 \times (-5) = 3 \times (5 + (-5)) = 3 \times 0 = 0$ So $3 \times (-5) = -15$ (the additive inverse of $15$ ).

Now consider $(-3) \times 5$ : $(-3) \times 5 + 3 \times 5 = ((-3) + 3) \times 5 = 0 \times 5 = 0$ So $(-3) \times 5 = -15$ .

Finally, $(-3) \times (-5)$ . Add $(-3) \times 5$ to it and factor out the common $(-3)$ : $(-3) \times (-5) + (-3) \times 5 = (-3) \times ((-5) + 5) = (-3) \times 0 = 0$ Since $(-3) \times 5 = -15$ , substitute: $(-3) \times (-5) + (-15) = 0 \implies (-3) \times (-5) = 15$

Why this step? We used the distributive property (factoring out $(-3)$ ) and the fact that any number times zero is zero. The negative-negative case is forced to be positive to keep the sum equal to $0$ .

Division

Rule: $a \div b = a \times \frac{1}{b}$ (multiply by reciprocal)

Sign rules (same as multiplication):

$(+) \div (+) = (+)$
$(+) \div (-) = (-)$
$(-) \div (+) = (-)$
$(-) \div (-) = (+)$

Why this step? Division undoes multiplication. If $(-4) \times (-5) = 20$ , then $20 \div (-4) = -5$ and $20 \div (-5) = -4$ must hold.

Critical: Division by zero is undefined. Why? No number $x$ satisfies $0 \times x = 5$ .

Absolute Value

Why this definition? Distance is always non-negative. If $a$ is already positive or zero, $|a| = a$ . If $a$ is negative, we flip the sign to get the positive distance: $|-5| = -(-5) = 5$ .

Deriving the triangle inequality:

Geometric intuition: If you walk $|a|$ steps, then $|b|$ steps, you travel at most $|a| + |b|$ total. If some steps backtrack, your final distance from start is less.

Algebraic proof (clean version):

A useful fact for any real $x$ : $-|x| \le x \le |x|$ (a number is squeezed between the negative and positive of its own magnitude).

Apply this to $a$ and $b$ : $-|a| \le a \le |a|, \qquad -|b| \le b \le |b|$

Add the two chains of inequalities term by term: $-\left(|a| + |b|\right) \le a + b \le |a| + |b|$

This says $a + b$ lies between $-(|a|+|b|)$ and $+(|a|+|b|)$ . By definition of absolute value, " $x$ lies between $-M$ and $M$ " is exactly " $|x| \le M$ ". Therefore: $|a + b| \le |a| + |b| \qquad \blacksquare$

Why this step? The key is that $|a+b|$ measures how far $a+b$ is from zero, and adding the two "squeeze" bounds gives us the largest possible distance. Equality holds only when $a$ and $b$ have the same sign (both move in the same direction).

Properties and Laws

Why these matter? They let us rearrange and simplify expressions confidently. For example, $3 \times (4 + 5)$ can be computed as $3 \times 4 + 3 \times 5$ (distributive), which is easier.

Common Mistakes

Common mistake Mistake 2:

$|a + b| = |a| + |b|$ always Wrong idea: $|-3 + 5| = |-3| + |5| = 3 + 5 = 8$

Why it feels right: Absolute value "removes negatives," so students distribute it like multiplication.

The fix: Absolute value applies to the result, not the parts. $|-3 + 5| = |2| = 2 \ne 8$ Triangle inequality says $|a+b| \le |a| + |b|$ (often strict). Equality holds only when $a$ and $b$ have the same sign (both move in the same direction).

Correct: $|3 + 5| = |8| = 8 = |3| + |5|$ ✓ (same sign)
Counterexample: $|-4 + 7| = |3| = 3 < |-4| + |7| = 4 + 7 = 11$

Active Recall Practice

Recall Feynman Explanation (for a 12-year-old)

Imagine you're walking along a straight path with a big zero painted in the middle. If you walk to the right, you're going into positive numbers—1, 2, 3. If you walk to the left, you're going into negative numbers—like walking backward: -1, -2, -3. Adding is just walking more steps. If I say "add 5," you walk 5 steps right. If I say "add -5," you walk 5 steps left (because the negative means "the opposite direction").

Subtracting is the same as adding the opposite. "Subtract 5" means "add -5" (walk left 5). "Subtract -5" means "add 5" (walk right 5)—that's why "minus a minus" becomes a plus!

Multiplying is like doing the same walk multiple times. 3 times 4 means "walk right 4 steps, three times" → you end up at 12. But 3 times -4 means "walk left 4 steps, three times" → you end up at -12. And -3 times -4? That's weird: it means "remove 3 groups of walking left 4 steps"—so you actually gain 12 steps to the right. Like erasing debt!

Absolute value is "how many steps away from zero are you?" Doesn't matter if you went left or right. $|7|$ is 7 steps, $|-7|$ is also 7 steps. It's the distance.

That's all integers are—a number line, some walking, and distance. Once you see it, all the rules make sense!

Connections

2.5.01-Natural-numbers-and-place-value — Integers extend naturals with negatives
2.5.03-Rational-numbers — Next step: integers divided by integers (fractions)
3.2.01-Linear-equations-in-one-variable — Solving $ax + b = c$ uses integer operations
4.1.02-Coordinate-plane — Number line becomes 2D: $x$ and $y$ axes are integer lines
6.3.01-Vectors-in-2D — Vector addition is like integer addition on each axis
2.1.05-Inequalities — Ordering on the number line: $a < b$ means $a$ is left of $b$

Flashcards

#flashcards/maths

What is the set of integers?

\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}

, all positive and negative whole numbers plus zero

On the number line, what does adding a negative number mean?

Moving left (in the negative direction)

Why does

(-3) + (-5) = -8

not

= +8

Both numbers move left, so distances add in the negative direction; "two negatives make positive" only applies to multiplication/division

Convert subtraction to addition:

7 - (-4) = ?

7 - (-4) = 7 + 4 = 11

(subtracting a negative is adding a positive)

Why is

(-2) \times (-3) = +6

Removing 2 groups of "3 units of debt" means gaining 6; formally, factoring out

(-2)

from

(-2)\times(-3) + (-2)\times 3 = (-2)\times 0 = 0

forces it to be

+6

What is

|a|

geometrically?

The distance from

a

to 0 on the number line (always non-negative)

State the triangle inequality for absolute value

|a + b| \le |a| + |b|

(total distance from zero is at most the sum of individual distances)

a < 0

, what is

|a|

|a| = -a

(flip the sign to get the positive distance)

Why is division by zero undefined?

No number

x

satisfies

0 \times x = a

for

a \ne 0

; the operation has no consistent result

What does the distributive law state?

a \times (b + c) = a \times b + a \times c

(multiplication distributes over addition)

True or false:

|a + b| = |a| + |b|

for all integers

False. Equality holds only when

a

and

b

have the same sign; otherwise

|a+b| < |a| + |b|

(triangle inequality)

Which operation has the identity element 0?

Addition (

a + 0 = a

)

Which operation has the identity element 1?

Multiplication (

a \times 1 = a

)

What is the additive inverse of

a

-a

(because

a + (-a) = 0

)

Why can't you distribute absolute value:

|a+b| \ne |a| + |b|

Absolute value applies to the final result after addition; if

a

and

b

point in opposite directions, they partially cancel, reducing the distance from zero

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, integers ka matlab hai poora whole number universe—jismein positive numbers, zero, aur negative numbers sab shaamil hote hain, jaise $\{...,-2,-1,0,1,2,...\}$ . Iski zaroorat isliye padi kyunki sirf natural numbers ( $1,2,3...$ ) se hum $3-5$ jaisa subtraction handle nahi kar sakte the. Toh negatives add karke maths ka system complete ho gaya, taaki har subtraction ka answer nikal sake. Real life mein bhi yeh kaam aate hain—jaise debt (udhaar), zero se neeche temperature, ya kisi starting point se left ki position dikhaana.

Ab sabse pyaari cheez hai number line—ek horizontal line socho jismein beech mein zero hai, right taraf positive aur left taraf negative. Isse arithmetic ek movement ban jaati hai: positive add karo toh right jao, negative add karo toh left jao. Isi liye $3+(-5)$ ka matlab hai "3 se shuru karo, 5 steps left jao, toh -2 pe pahucho". Subtraction ko toh hum addition mein hi badal dete hain— $a-b = a+(-b)$ —yaani "minus karna matlab opposite add karna". Aur absolute value ka matlab bas itna hai ki koi number zero se kitni door hai, direction ki tension nahi.

Sign rules bhi ratne ki cheez nahi hain, samajhne ki hain. Same sign wale add karo toh magnitudes jud jaate hain (dono ek hi direction mein move kar rahe hain). Opposite signs mein ek dusre ko cancel karte hain aur jo bada hai uski taraf answer jaata hai. Aur woh famous rule ki $(-)\times(-)=(+)$ —iske peeche logic hai "debt ko hataana": agar tumhaare 3 debts hain $5 each ke, aur tum unhe remove kar do, toh tumhe$ 15 ka fayda ho jaata hai. Yeh intuition pakad lo toh number theory ki poori nemv strong ho jaati hai, kyunki aage har topic inhi basics pe khadi hoti hai.

Integers — operations, number line, absolute value

What Are Integers?

The Number Line

Operations on Integers

Addition

Subtraction

Multiplication

Division

Absolute Value

Properties and Laws

Common Mistakes

Active Recall Practice

Connections

Flashcards

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Number Theory (Intermediate)

Connections