Integers — operations, number line, absolute value

2.5.2 · Maths › Number Theory (Intermediate)

What Are Integers?

Integers kyun matter karte hain: Natural numbers $\{1,2,3,\dots \}$ $3-5$ jaisi subtraction handle nahi kar sakti. Integers system ko complete karte hain taaki har subtraction ka answer ho.

The Number Line

Key features:

• Origin: 0 center hai
• Direction: Right = positive, Left = negative
• Distance: Points ke beech ke steps
• Ordering: $a<b$ ka matlab "$a$, $b$ ke left hai"

Operations on Integers

Addition

Movement se rule:

• Positive number add karna → right move karo
• Negative number add karna → left move karo

Yeh step kyun? Har "left by 5 move" exactly wohi hai jo $-5$ add karna mean karta hai. Number line is rule ko visual banata hai.

Sign rules ki derivation:

• Same signs: Agar dono positive ya dono negative hain, to magnitudes add karo aur sign same rakho.
  Kyun? Same direction mein move karne se distance accumulate hoti hai.
  $(-a)+(-b)=-(a+b)\text{dono left move karte hain}$

• Opposite signs: Chhoti magnitude ko badi se subtract karo, badi wali ka sign lo. Agar magnitudes equal hain, to result 0 hai.
  Kyun? Ek movement doosre ko partially cancel karta hai. "Winner" final side determine karta hai; tie hone par exactly 0 pe land karte ho.
  $a+(-b)=\{\begin{array}{ll}a-b& \text{if }a>b\\ 0& \text{if }a=b\\ -(b-a)& \text{if }b>a\end{array}$

Subtraction

Addition ke roop mein likhna: $a-b=a+(-b)$

Kyun? Subtract karna opposite add karne jaisa hai. "5 le lo" = "negative 5 add karo" (5 left move karo).

Yeh step kyun? Addition mein convert karne se hum ek hi addition rule use kar sakte hain. "Negative subtract karna" "positive add karna" ban jaata hai kyunki $-(-6)=6$.

Multiplication

Repeated addition se rule: $a\times b=\underset{b\text{ times}}{\underbrace{a+a+\cdots +a}}\text{(if }b>0\text{)}$

Sign rules:

• (+) × (+) = (+): Positives add karna positive rehta hai
• (+) × (−) = (−): Negatives add karna (opposite direction) negative deta hai
• (−) × (+) = (−): Same as above (commutative)
• (−) × (−) = (+): KYUN? "Debt hatao" socho. Agar tum 3 debts of $5 each remove karo, to tumhe $15 milte hain.

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

Additive inverse property se start karo: $a+(-a)=0$ kisi bhi $a$ ke liye.

Maano $a=3$ aur $b=5$: $3\times 5+3\times (-5)=3\times (5+(-5))=3\times 0=0$ To $3\times (-5)=-15$ ($15$ ka additive inverse).

Ab $(-3)\times 5$ socho: $(-3)\times 5+3\times 5=((-3)+3)\times 5=0\times 5=0$ To $(-3)\times 5=-15$.

Aakhir mein, $(-3)\times (-5)$. Isme $(-3)\times 5$ add karo aur common $(-3)$ factor out karo: $(-3)\times (-5)+(-3)\times 5=(-3)\times ((-5)+5)=(-3)\times 0=0$ Kyunki $(-3)\times 5=-15$, substitute karo: $(-3)\times (-5)+(-15)=0⟹(-3)\times (-5)=15$

Yeh step kyun? Humne distributive property (common $(-3)$ factor out karna) use ki aur yeh fact use kiya ki koi bhi number zero se multiply hone par zero deta hai. Negative-negative case forced hai positive hone ke liye taaki sum $0$ ke equal rahe.

Division

Rule: $a÷b=a\times \frac{1}{b}$ (reciprocal se multiply karo)

Sign rules (multiplication jaisi hi):

• $(+)÷(+)=(+)$
• $(+)÷(-)=(-)$
• $(-)÷(+)=(-)$
• $(-)÷(-)=(+)$

Yeh step kyun? Division multiplication ko undo karta hai. Agar $(-4)\times (-5)=20$, to $20÷(-4)=-5$ aur $20÷(-5)=-4$ hold karna chahiye.

Critical: Division by zero undefined hai. Kyun? Koi number $x$ satisfy nahi karta $0\times x=5$.

Absolute Value

Yeh definition kyun? Distance hamesha non-negative hoti hai. Agar $a$ already positive ya zero hai, to $|a|=a$. Agar $a$ negative hai, to hum sign flip karte hain positive distance paane ke liye: $|-5|=-(-5)=5$.

Triangle inequality derive karna:

Geometric intuition: Agar tum $|a|$ steps chalte ho, phir $|b|$ steps, to tum zyada se zyada $|a|+|b|$ total chalte ho. Agar kuch steps backtrack karte hain, to start se tumhari final distance kam hoti hai.

Algebraic proof (clean version):

Kisi bhi real $x$ ke liye ek useful fact: $-|x|\le x\le |x|$ (ek number apni khud ki magnitude ke negative aur positive ke beech squeezed hota hai).

Isse $a$ aur $b$ par apply karo: $-|a|\le a\le |a|,-|b|\le b\le |b|$

Dono chains of inequalities ko term by term add karo: $-\left(|a|+|b|\right)\le a+b\le |a|+|b|$

Iska matlab hai $a+b$, $-(|a|+|b|)$ aur $+(|a|+|b|)$ ke beech hai. Absolute value ki definition se, "$x$, $-M$ aur $M$ ke beech hai" exactly "$|x|\le M$" hai. Isliye: $|a+b|\le |a|+|b|■$

Yeh step kyun? Key yeh hai ki $|a+b|$ measure karta hai $a+b$ zero se kitna door hai, aur dono "squeeze" bounds add karne se hume maximum possible distance milti hai. Equality tabhi hoti hai jab $a$ aur $b$ ka same sign ho (dono same direction mein chalte hain).

Properties and Laws

Ye kyun matter karte hain? Ye hume expressions confidently rearrange aur simplify karne dete hain. For example, $3\times (4+5)$ ko $3\times 4+3\times 5$ (distributive) ke roop mein compute kiya ja sakta hai, jo aasaan hai.

Common Mistakes

Active Recall Practice

Connections

• 2.5.01-Natural-numbers-and-place-value — Integers, naturals ko negatives ke saath extend karte hain
• 2.5.03-Rational-numbers — Agla step: integers divided by integers (fractions)
• 3.2.01-Linear-equations-in-one-variable — $ax+b=c$ solve karna integer operations use karta hai
• 4.1.02-Coordinate-plane — Number line 2D ban jaata hai: $x$ aur $y$ axes integer lines hain
• 6.3.01-Vectors-in-2D — Vector addition har axis par integer addition jaisi hai
• 2.1.05-Inequalities — Number line par ordering: $a<b$ matlab $a$, $b$ ke left hai

Flashcards

#flashcards/maths

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps