Integers | Speed Notes

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Whole Numbers The numbers 1,2, 3, ……which we use for counting are known as natural numbers. If you add 1 to a natural number, we get its successor. If you subtract 1 from a natural number, you get its predecessor. (Scroll down to continue …)

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Integers

Definition

Integers are the set of whole numbers that include positive numbers, negative numbers, and zero. The set of integers can be represented as: Integers={…,−3,−2,−1,0,1,2,3,…}Integers={…,−3,−2,−1,0,1,2,3,…}

Key Properties of Integers

1. Closure Properties:
   • Addition: The sum of any two integers is an integer.
     • Examples:
       • 2+3=52+3=5
       • −1+4=3−1+4=3
       • −2+(−3)=−5−2+(−3)=−5
   • Subtraction: The difference between any two integers is an integer.
     • Examples:
       • 5−3=25−3=2
       • −2−1=−3−2−1=−3
       • 0−(−4)=40−(−4)=4
   • Multiplication: The product of any two integers is an integer.
     • Examples:
       • 3×2=63×2=6
       • −4×5=−20−4×5=−20
       • −3×−2=6−3×−2=6
2. Identity Elements:
   • Additive Identity: The integer 0 is the identity element for addition.
     • Examples:
       • 7+0=77+0=7
       • −5+0=−5−5+0=−5
       • 0+0=00+0=0
   • Multiplicative Identity: The integer 1 is the identity element for multiplication.
     • Examples:
       • 4×1=44×1=4
       • −3×1=−3−3×1=−3
       • 0×1=00×1=0
3. Inverse Elements:
   • Additive Inverse: For every integer a, there exists an integer −a such that a+(−a)=a+−a=0.
     • Examples:
       • The additive inverse of 5 is -5: 5+(−5)=5+−5=0
       • The additive inverse of -3 is 3: −3+3=0
       • The additive inverse of 0 is 0: 0+0=0
   • Multiplicative Inverse: Integers do not have multiplicative inverses within the set of integers (except for 1 and -1).
4. Commutative and Associative Properties:
   • Commutative Property:
     • Addition: a+b = b+a
       • Examples:
         • 2+3=3+2
         • −1+4 = 4+(−1) = 4-1 = 3
         • 0+5 = 5+0 = 5
     • Multiplication: a×b=b×a
       • Examples:
         • 3×4 = 4×3 = 12
         • −2×1 = 1×−2 = -2
         • 0×5 = 5×0 = 0
   • Associative Property:
     • Addition: (a+b)+c = a+(b+c) = (a+c)+b
       • Examples:
         • (1+2)+3 = 1+(2+3) = (1+3)+2
         • [0+(−4)]+2 = 0+[−4+2] = [(0+2)+(-4)]
         • [-2+(−3)]+(-1) = -2+[−3+(-1)] = [-2+(−1)]+(-3)
     • Multiplication: (a×b)×c=a×(b×c)(a×c)×b
       • Examples:
         • (2×3)×4 = 2×(3×4) = (2×4)×3
         • (0×−1)×5 = 0×(−1×5) = (0×5)×−1
         • (−2×3)×−1 = −2×(3×−1) = (−2×-1)×3
5. Distributive Property:
   • Multiplication distributes over addition:
     • Example: a×(b+c)=(a×b)+(a×c) Or a×(b+c)=a×b+a×c
       • Examples:
         • 2×(3+4) = (2×3)+(2×4) = 6+12 = 14 Or (2×7) = 14
         • −3×(1+2) = (−3×1)+(−3×2) = -3-6 = -9 Or −3×3 = −9
         • 0×(5+7) = (0×5)+(0×7) = 0×(5+7) = 0×5+0×7 = 0+0 =0

Ordering of Integers

• Integers can be ordered on a number line, where:
  • Negative integers are to the left of 0.
  • Positive integers are to the right of 0.
• Examples of ordering:
  • …−3<−2<−1<0<1<2<3−3<−2<−1<0<1<2<3…
  • −5,−2,0,4,3−5,−2,0,4,3 arranged in order: −5<−2<0<3<4−5<−2<0<3<4

Absolute Value

• The absolute value of an integer is its distance from zero on the number line, regardless of direction.
  • Notation: ∣a∣∣a∣
  • Examples:
    • ∣3∣=3∣3∣=3
    • ∣−3∣=3∣−3∣=3
    • ∣0∣=0∣0∣=0

Conclusion

Understanding integers and their properties is fundamental in mathematics. They play a critical role in various areas, including algebra, number theory, and real-world applications. Mastery of integer operations is essential for higher-level mathematics.