ARITHMETIC EXPRESSIONS

Arithmetic expressions mathematical phrases hote hain jo numbers aur operations (+, -, ×, ÷) se bante hain. Har expression ki ek =value= hoti hai.

• Definition: Ek mathematical phrase jisme numbers aur operations hote hain, jaise 13 + 2, 20 - 4, 12 × 5, 18 ÷ 3.
• Value: Har arithmetic expression ki ek single numerical value hoti hai. Jaise, 13 + 2 ki value 15 hai.
• Reading Expressions:
• 13 + 2: "13 plus 2" ya "the sum of 13 and 2".
• 20 - 4: "20 minus 4" ya "the difference of 20 and 4".
• 12 × 5: "12 times 5" ya "the product of 12 and 5".
• 18 ÷ 3: "18 divided by 3" ya "the quotient of 18 and 3".
• Equality Sign (=): Ye expression aur uski value ke beech ka relationship batata hai. Example: 13 + 2 = 15.
• Same Value, Different Expressions: Ek hi number ko alag-alag expressions se represent kiya ja sakta hai. Jaise, 12 ko 10 + 2, 15 - 3, 3 × 4, 24 ÷ 2 se represent kar sakte hain.

Yaad rakho, ek expression mein sirf numbers aur basic arithmetic operations hote hain. Variables (jaise x, y) algebraic expressions mein aate hain, jo tum higher classes mein padhoge.

Example: Mallika har din lunch par ₹25 kharch karti hai. Monday se Friday tak usne kitna kharch kiya?

• Days: 5 (Monday to Friday)
• Daily expense: ₹25
• Expression: 5 × 25
• Value: 125

Jaise hum numbers ko ( = ), ( < ), ( > ) signs se compare karte hain, waise hi expressions ko bhi unki values ke basis par compare kar sakte hain.

• Comparison Rule: Pehle har expression ki value find karo, fir un values ko compare karo.
• Signs Used: ( = ) (equal to), ( < ) (less than), ( > ) (greater than).

Example 1: 10 + 2 aur 7 + 1 ko compare karo.

• 10 + 2 = 12
• 7 + 1 = 8
• Kyunki 12 > 8, toh 10 + 2 > 7 + 1.

Example 2: 13 - 2 aur 4 × 3 ko compare karo.

• 13 - 2 = 11
• 4 × 3 = 12
• Kyunki 11 < 12, toh 13 - 2 < 4 × 3.

Tricky Comparisons (without direct calculation): Kabhi-kabhi expressions ko bina poori value calculate kiye bhi compare kiya ja sakta hai, khaaskar jab numbers mein chhota sa difference ho.

• Scenario 1: Addition
• Compare 1023 + 125 aur 1022 + 128.
• 1023 (Raja ke paas) 1022 (Joy ke paas) se 1 zyada hai.
• 125 (Raja ko mile) 128 (Joy ko mile) se 3 kam hain.
• So, Raja ke paas shuru mein 1 zyada tha, but Joy ko 3 zyada mile. Net effect: Joy ke paas 3 - 1 = 2 zyada hain.
• Iska matlab 1023 + 125 < 1022 + 128.

• Scenario 2: Subtraction
• Compare 113 - 25 aur 112 - 24.
• 113 (Raja ke paas) 112 (Joy ke paas) se 1 zyada hai.
• 25 (Raja ne khoye) 24 (Joy ne khoye) se 1 zyada hain.
• Raja ke paas shuru mein 1 zyada tha, aur usne 1 zyada hi khoya. So, dono ke paas ab equal hain.
• Iska matlab 113 - 25 = 112 - 24.

Exam mein aise questions aate hain jahan direct calculation ki jagah logical reasoning use karni hoti hai.

Jab ek expression mein multiple operations hote hain, toh confusion ho sakti hai ki pehle kaun sa operation perform karein. Is confusion ko avoid karne ke liye brackets ( ) use kiye jaate hain.

• Role of Brackets: Brackets ( ) specify karte hain ki kaun sa part pehle evaluate karna hai.
• Order of Operations: Jab bhi brackets hon, toh sabse pehle brackets ke andar wale expression ko solve karte hain.

Example: 30 + 5 × 4

• Confusion: Kya pehle 30 + 5 karein ya 5 × 4?
• Agar (30 + 5) × 4 = 35 × 4 = 140 (Galat)
• Agar 30 + (5 × 4) = 30 + 20 = 50 (Sahi, as per BODMAS/PEMDAS rule)
• Solution: Brackets use karke order clear kiya jaata hai: 30 + (5 × 4).

BODMAS/PEMDAS Rule (Order of Operations): Ye rule batata hai ki kis order mein operations ko perform karna hai:

1. Brackets (Parentheses)
2. Orders (Exponents/Powers)
3. Division and Multiplication (Left to right)
4. Addition and Subtraction (Left to right)

Example: 10 + (8 ÷ 2) - 3 × 2

1. Brackets: 10 + 4 - 3 × 2
2. Multiplication: 10 + 4 - 6
3. Addition/Subtraction (left to right): 14 - 6 = 8

Brackets ka sahi use complex expressions ko clear aur correct banata hai.

Terms ek expression ke parts hote hain jo + sign se separate hote hain. Subtraction ko bhi addition of negative number ki tarah treat kiya ja sakta hai.

• Definition: Expression ke woh parts jo + sign se alag hote hain, unhe terms kehte hain.
• Subtraction as Addition: a - b ko a + (-b) likha ja sakta hai. Isse a aur -b terms ban jaate hain.

Example 1: 12 + 7

• Terms: 12 aur 7.

Example 2: 83 - 14

• Isse 83 + (-14) likh sakte hain.
• Terms: 83 aur -14.

Example 3: 30 + 5 × 4

• Pehle 5 × 4 ko evaluate karo (20).
• Expression banega 30 + 20.
• Terms: 30 aur 20.

Complex Expressions mein Terms: Jab brackets hote hain, toh brackets ke andar ka poora expression ek single term ban jaata hai, jab tak use solve na kar liya jaaye.

Example: (10 + 5) - 3

• Pehle (10 + 5) ko solve karo: 15.
• Expression banega 15 - 3 ya 15 + (-3).
• Terms: 15 aur -3.

Terms ko sahi se identify karna Commutative aur Associative properties apply karne mein help karta hai.

Note: Multiplication aur Division terms ko separate nahi karte. 5 × 4 ek single term hai.

Ye properties batati hain ki addition operations mein terms ke order ya grouping se result change nahi hota.

1. Commutative Property of Addition

• Meaning: Jab do numbers ko add karte hain, toh unka order change karne se sum same rehta hai.
• Rule: a + b = b + a
• Example: 5 + 3 = 8 aur 3 + 5 = 8. Dono same hain.
• Real-life analogy: Shoes pehle pehno ya socks pehle pehno, order matter karta hai. Lekin agar tum hat aur shoes pehen rahe ho, toh order matter nahi karta. [IMAGE: order_matters_everyday_examples_fig25]

2. Associative Property of Addition

• Meaning: Jab teen ya teen se zyada numbers ko add karte hain, toh unki grouping change karne se sum same rehta hai.
• Rule: (a + b) + c = a + (b + c)
• Example: (2 + 3) + 4 = 5 + 4 = 9 aur 2 + (3 + 4) = 2 + 7 = 9. Dono same hain.
• Note: Ye properties sirf addition aur multiplication par apply hoti hain. Subtraction aur Division par nahi.

Why these are important for expressions?

• Ye properties expressions ko simplify karne mein help karti hain.
• Agar ek expression mein sirf addition ke terms hain, toh unhe kisi bhi order mein add kar sakte ho ya kisi bhi tarah group kar sakte ho, final answer same hi aayega.

Example: 28 - 7 + 8

• Isse 28 + (-7) + 8 likh sakte hain.
• Commutative: 28 + 8 + (-7) = 36 - 7 = 29
• Associative: (28 + (-7)) + 8 = 21 + 8 = 29

ya 28 + ((-7) + 8) = 28 + 1 = 29

In properties ko samajhne se calculations easy ho jaati hain.

Caution: Subtraction aur division ke liye ye properties apply nahi hoti. (a - b) - c ≠ a - (b - c) aur (a ÷ b) ÷ c ≠ a ÷ (b ÷ c).

Brackets ko remove karte waqt, uske pehle wale sign ka dhyaan rakhna bahut zaroori hai. Isse expression ki value change ho sakti hai.

Rule 1: Brackets ke pehle + sign ho

• Agar bracket ke pehle + sign hai, toh bracket remove karne par andar ke terms ke signs change nahi hote.
• Rule: a + (b + c) = a + b + c
• Example: 10 + (5 + 2) = 10 + 5 + 2 = 17
• Example: 10 + (5 - 2) = 10 + 5 - 2 = 13

Rule 2: Brackets ke pehle - sign ho

• Agar bracket ke pehle - sign hai, toh bracket remove karne par andar ke sabhi terms ke signs reverse ho jaate hain. + ban jaata hai - aur - ban jaata hai +.
• Rule: a - (b + c) = a - b - c
• Rule: a - (b - c) = a - b + c
• Example: 10 - (5 + 2) = 10 - 5 - 2 = 3
• Example: 10 - (5 - 2) = 10 - 5 + 2 = 7

Rule 3: Brackets ke pehle koi sign na ho (multiplication implied)

• Agar bracket ke pehle koi sign nahi hai, toh multiplication implied hota hai. Is case mein Distributive Property apply hoti hai (next topic).
• Example: 3(5 + 2) matlab 3 × (5 + 2).

Ye rules bahut critical hain complex expressions ko simplify karne ke liye.

Nested Brackets: Jab ek bracket ke andar doosra bracket ho, toh sabse andar wale bracket se shuru karte hain aur bahar ki taraf solve karte hain.

• Types of brackets: () (parentheses), {} (braces), [] (square brackets).
• Order: [ { ( ) } ]

Example: 10 - [ 5 + { 3 - (2 - 1) } ]

1. Innermost (2 - 1) = 1
2. Next { 3 - 1 } = 2
3. Next [ 5 + 2 ] = 7
4. Finally 10 - 7 = 3

Distributive property batati hai ki multiplication addition ya subtraction par kaise distribute hota hai. Ye property calculations ko simplify karne mein bahut helpful hai.

• Meaning: Jab ek number ko ek sum ya difference se multiply karte hain, toh woh number sum/difference ke har term se individually multiply hota hai.

Rule 1: Multiplication over Addition

• Rule: a × (b + c) = (a × b) + (a × c)
• Example: 3 × (5 + 2)
• Direct: 3 × 7 = 21
• Distributive: (3 × 5) + (3 × 2) = 15 + 6 = 21

Rule 2: Multiplication over Subtraction

• Rule: a × (b - c) = (a × b) - (a × c)
• Example: 4 × (10 - 3)
• Direct: 4 × 7 = 28
• Distributive: (4 × 10) - (4 × 3) = 40 - 12 = 28

Visual Analogy: Imagine ek window hai jismein grills lage hain. [IMAGE: window_measurements_fig27]

• Agar window ki width a hai aur uske andar do sections b aur c hain, toh total area a × (b + c) hoga.
• Ye area do chhote areas a × b aur a × c ke sum ke barabar hoga.

Why is this property important?

• Mental math mein helpful. Jaise 103 × 7 ko (100 + 3) × 7 = 100 × 7 + 3 × 7 = 700 + 21 = 721 calculate kar sakte hain.
• Algebraic expressions ko simplify karne mein foundation provide karta hai.

Distributive property ek bahut powerful tool hai calculations ko easy banane ke liye.

Note: Ye property sirf multiplication ke liye hai jo addition ya subtraction par distribute hota hai. Addition multiplication par distribute nahi hota: a + (b × c) ≠ (a + b) × (a + c).