Page 25 - Math_Genius_V1.0_C7_Flipbook

D:\Surender Prajapati\CBSE_ICSE_Book_New\CBSE\Grade-7\Math_Genius-7\Open_File\01_Chapter\01_Chapter
\ 15-Nov-2024 Surender Prajapati Proof-6 Reader’s Sign _______________________ Date __________

Encapsulate
math
INTEGERS

Negative Numbers like –4, –3, –2, –1, etc. Whole Numbers

Zero Positive Numbers like 1, 2, 3, etc.

Operations on Integers

Addition Subtraction

• 3 + 2 = 5 • 2 + (–1) = 1 • 4 – 3 = 1 • 5 – (–2) = 7
• (–3) + 4 = 1 • –2 + (–3) = –5 • (–2) – (–1) = –1 • (–4) – 3 = –7

Properties Properties
• Closure: a + b = integer, e.g., 4 + (–2) = 2 • Closure: a – b = integer, e.g., 3 – 2 = 1
• Commutative: a + b = b + a, e.g., (–3) + 2 = 2 + (–3) • Not commutative: a – b ≠ b – a,
= –1 e.g., 4 – (–2) ≠ (–2) – 4
• Associative: a + (b + c) = (a + b) + c, • Not associative: a – (b – c) ≠ (a – b) – c,
e.g., (–4) + (2 + 3) = (–4 + 2) + 3 = 1 e.g., 3 – (2 – 1) ≠ (3 – 2) – 1
• Additive inverse: a + (–a) = (–a) + a = 0, • a – b = a + (additive inverse of b),
e.g., 2 + (–2) = 0 = (–2) +2 e.g., 4 – (–2) = 4 + additive inverse of (–2)
• Additive identity: a + 0 = 0 + a = a, = 4 + (+2) = 6
e.g., (–5) + 0 = –5 = 0 + (–5) • Property of zero: a – 0 = a, e.g., 3 – 0 = 3

Multiplication Division

• 2 × 3 = 6 • 2 × (–1) = –2 • 4 ÷ 2 = 2 • –6 ÷ 3 = –2
• (–3) × 4 = –12 • (–2) × (–3) = 6 • (–8) ÷ (–4) = 2 • 12 ÷ (–3) = –4

Properties Properties
• Closure: a × b = integer, e.g., –3 × 2 = –6 • Not closure: a ÷ b = not necessarily an integer,
• Commutative: a × b = b × a, e.g., –3 × 2 = 2 × (–3) 1
• Associative: a × (b × c) = (a × b) × c, e.g., 1 ÷ 2 = 2 = 0.5
e.g., –2 × (3 × 4) = {(–2) × 3} × 4 • Not commutative: a ÷ b ≠ b ÷ a,
• Multiplicative identity: a × 1 = a = 1 × a, e.g., –3 ÷ 2 ≠ 2 ÷ (–3)
e.g., (–4) × 1 = –4 = 1 × (–4) • Not associative: a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c,
• Property of zero: a × 0 = 0 = 0 × a, e.g., (–3) ÷ (3 ÷ 2) ≠ {(–3) ÷ 3} ÷ 2
e.g., (–5) × 0 = 0 = 0 × (–5)
1 1 • Property of 1: a ÷ 1 = a,
• Multiplicative inverse: a × = 1 = × a (a ≠ 0), e.g., –3 ÷ 1 = –3
a a
1 • Property of zero: a ÷ 0 = meaningless, 0 ÷ a = a,
e.g., − ( )× = 1
2
− ( ) 2 e.g., 0 ÷ 1 = 1
• Distributive: a × (b + c) = (a × b) + (a × c), • For any integer a, a ÷ a = 1,
e.g., –2 × (3 + 4) = (–2) × 3 + (–2) × 4 e.g., –3 ÷ (–3) = 1

23 Integers