IN: Extended Concept

Integers Extended Concept

1. Definition:

  • Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. The set of integers is denoted by \mathbb{Z} and includes: \{\ldots,-3,-2,-1,0,1,2,3,\ldots\}.

2. Properties of Integers:

  • Addition and Subtraction:
    • The sum or difference of two integers is always an integer.
    • Example: 5+(-3)=2, -4-6=-10.
  • Multiplication:
    • The product of two integers is always an integer.
    • Example: 3\times(-4)=-12, -2\times(-5)=10.
  • Division:
    • The quotient of two integers is not always an integer.
    • Example: 6\div3=2 (integer), but 7\div2=3.5 (not an integer).

3. Even and Odd Integers:

  • Even Integers: Divisible by 2. Examples: -4,-2,0,2,4,\ldots
  • Odd Integers: Not divisible by 2. Examples: -3,-1,1,3,5,\ldots

4. Positive and Negative Integers:

  • Positive Integers: Greater than zero. Examples: 1,2,3,\ldots
  • Negative Integers: Less than zero. Examples: -1,-2,-3,\ldots

5. Zero (0):

  • Zero is an integer that is neither positive nor negative.
  • It is an even number.

6. Absolute Value:

  • The absolute value of an integer is its distance from zero on the number line, regardless of direction.
  • Denoted by |a|.
  • Example: |3|=3, |-3|=3.

7. Number Line:

  • Integers are represented on a number line with zero at the center, positive integers to the right, and negative integers to the left.

8. Properties of Operations:

  • Commutative Property:
    • Addition: a+b=b+a
    • Multiplication: a\times b=b\times a
  • Associative Property:
    • Addition: (a+b)+c=a+(b+c)
    • Multiplication: (a\times b)\times c=a\times(b\times c)
  • Distributive Property:
    • a\times(b+c)=(a\times b)+(a\times c)

9. Practical Applications:

  • GRE/GMAT Problems: Integers are used in various types of quantitative questions, including algebra, number properties, and word problems. Understanding their properties helps in solving such problems efficiently.

Practice Questions

1. If x and y are integers such that x is positive and y is negative, which of the following must be true?

  • A) x+y is positive
  • B) x\times y is positive
  • C) x-y is positive
  • D) x\div y is an integer

2. Which of the following expressions always results in an even number?

  • A) 3a+4b where a and b are integers
  • B) a^2+b^2 where a and b are even integers
  • C) 2a+3b where a and b are integers
  • D) a^2-b^2 where a and b are integers

3. If n is an integer, which of the following is true about n^2?

  • A) n^2 is always positive
  • B) n^2 is always even
  • C) n^2 is always odd
  • D) n^2 is always non-negative

4. Given that a and b are integers, which of the following statements is true?

  • A) a+b is always even
  • B) a\times b is always even
  • C) a-b is always odd
  • D) a\div b is not always an integer

5. If x is an even integer and y is an odd integer, which of the following is always true?

  • A) x+y is even
  • B) x-y is odd
  • C) x\times y is odd
  • D) x\div y is an integer

Answers:

  1. C) x-y is positive
  2. B) a^2+b^2 where a and b are even integers
  3. D) n^2 is always non-negative
  4. D) a\div b is not always an integer
  5. B) x-y is odd

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