Integers

Definition

Integers are a set of whole numbers that include both positive and negative values, as well as zero. They are essential in mathematics because they allow for operations like addition, subtraction, and multiplication without the need for fractions or decimals. Integers help in establishing relationships between numbers and are fundamental when discussing properties of real numbers and algebraic operations.

5 Must Know Facts For Your Next Test

1. Integers include all positive whole numbers, negative whole numbers, and zero.
2. The set of integers is often represented by the symbol 'Z', which comes from the German word 'Zahlen' meaning 'numbers'.
3. Integers do not include fractions or decimals; they are strictly whole numbers.
4. When performing arithmetic operations with integers, results may also be integers; for instance, adding two integers always yields another integer.
5. Integers are closed under addition, subtraction, and multiplication but not under division.

Review Questions

• How do integers relate to the concept of closure in mathematical operations?
  • Integers are closed under addition, subtraction, and multiplication, meaning that when you perform these operations on any two integers, the result is always an integer. For example, if you add -3 and 5, you get 2, which is also an integer. However, integers are not closed under division because dividing one integer by another can yield a non-integer result, such as 1 divided by 2.
• Discuss how the properties of integers support algebraic operations and provide examples.
  • The properties of integers facilitate various algebraic operations by ensuring consistent outcomes. For instance, the commutative property states that a + b = b + a; applying this to integers like 3 and -2 gives us 1 regardless of the order. The associative property further supports grouping in addition or multiplication without changing the outcome. These properties allow for more complex algebraic expressions to be simplified effectively.
• Evaluate the significance of absolute value in understanding integers within real number properties.
  • Absolute value plays a crucial role in understanding integers as it quantifies their distance from zero on the number line. This concept helps to simplify expressions and inequalities involving integers. For example, knowing that |-4| = 4 provides insights into comparing magnitudes regardless of sign. In algebraic contexts, it assists in solving equations and inequalities by removing negative signs, ensuring a clearer understanding of relationships between integer values.

Related terms

Whole Numbers: Whole numbers are non-negative integers, which include zero and all positive integers (0, 1, 2, 3, ...).

Rational Numbers: Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero.

Absolute Value: The absolute value of an integer is its distance from zero on the number line, regardless of direction, represented by |x|.