Absolute value - modulus — definition, number line interpretation

1.1.22 · Maths › Arithmetic & Number Systems

Overview

Kisi number ki absolute value (ya modulus) uski zero se distance measure karta hai number line par, direction ko ignore karte hue. Yeh har number ko uski non-negative magnitude mein badal deta hai.

Core concept

Yeh definition kyun?

• $x\ge 0$ ke liye: Number already apni zero se distance hai → $|x|=x$.
• $x<0$ ke liye: Number negative hai, lekin distance positive honi chahiye → hum ise negate karte hain: $|x|=-x$ (jo positive hai kyunki $x$ negative hai).

Number line interpretation

Number line humein ek geometric view deta hai:

$\text{Distance from }a\text{ to }b=|b-a|$

First principles se derivation:

1. Distance kya hai? Do points ke beech ka gap, hamesha non-negative.
2. Agar $b>a$ hai, toh gap $b-a$ hai (pehle se positive).
3. Agar $b<a$ hai, toh gap $a-b$ hai (hum subtraction reverse karte hain use positive rakhne ke liye).
4. Absolute value dono cases ko unify karta hai: $|b-a|$ sign ko automatically handle karta hai.

Subtract kyun karte hain? Kyunki line par position coordinates se di jaati hai; coordinates subtract karne se displacement milta hai, aur absolute value magnitude deta hai.

Properties (definition se derive ki gayi)

Property 1: Non-negativity

$|x|\ge 0\text{ for all }x\in \mathbb{R}$

Kyun? Definition ke anusar, hum ya toh ek non-negative number rakhte hain ya ek negative ko flip karte hain. Dono non-negative results produce karte hain.

Property 2: $|x|=0$ if and only if $x=0$

Kyun? Zero hi ek aisa number hai jo apne aap se zero distance par hai.

Property 3: $|-x|=|x|$

Derivation:

• Agar $x\ge 0$, toh $-x\le 0$, isliye $|-x|=-(-x)=x=|x|$. ✓
• Agar $x<0$, toh $-x>0$, isliye $|-x|=-x=|x|$. ✓

Geometric intuition: Zero ke across ek point ko reflect karne se uski zero se distance nahi badlti.

Property 4: Triangle inequality

$|x+y|\le |x|+|y|$

Yeh kyun hold karta hai? Chalne ke baare mein socho: agar tum $x$ meter chalte ho, phir $y$ meter chalte ho, toh start se end tak ki straight-line distance (yahi $|x+y|$ hai) kabhi bhi total path length ($|x|+|y|$) se zyada nahi ho sakti. Equality tab hoti hai jab tum seedhi line mein chalte ho (same direction); inequality tab hoti hai jab tum wapas mudke chalte ho.

Algebraic derivation (outline):

• $x,y$ ke signs ke aadhar par chaar cases consider karo.
• Har case mein, definitions expand karke $|x+y|\le |x|+|y|$ verify karo.

Worked examples

Common mistakes

Memory aids

Active recall practice

#flashcards/maths

Feynman technique

Connections

• Number line and ordering — absolute value number line ki geometry mein rooted hai
• Distance and metric spaces — $|x-y|$ $\mathbb{R}$ par standard metric define karta hai
• Inequalities — absolute value inequalities ($|x|<a$, $|x-c|<r$) intervals describe karte hain
• Complex numbers — modulus — $z=a+bi$ ke liye $|z|=\sqrt{a^2+b^2}$ tak extend hota hai
• Piecewise functions — absolute value sabse simple piecewise-defined function hai
• Triangle inequality — analysis, norms, aur metric spaces ke liye fundamental
• Solving absolute value equations — sign ke basis par case splitting ki technique

Summary: Absolute value $|x|$ zero se distance hai. Yeh hamesha non-negative, symmetric ($|-x|=|x|$) hota hai, aur triangle inequality satisfy karta hai. Number line par, $|b-a|$ $a$ aur $b$ ke beech distance deta hai. Absolute value involve karne wali equations sign ke basis par cases mein split hoti hain. Yeh concept inequalities, metrics, aur analysis ko underpin karta hai.

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