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Chapter 1: Problem 9

Describe each set of numbers. Odd integers

Short Answer

Expert verified
Odd integers are a subset of integers which leave a remainder of 1 when divided by 2. They can be represented in the form \(n = 2k + 1\), where 'n' and 'k' are integers. Examples include -5, -3, -1, 1, 3, 5, etc.

Step by step solution

01

Definition of Integers

Firstly, integers are a subset of whole numbers that include positive and negative numbers but no fractions or decimals. They can be represented on a number line. Examples: -2, -1, 0, 1, 2, etc.
02

Definition of Odd Integers

The next important concept is 'odd integers'. An odd integer is one that, when divided by 2, leaves a remainder of 1. To put it mathematically, if 'n' is a integer and can be represented in this form \(n = 2k + 1\) where 'k' is also an integer, then 'n' is an odd integer.
03

Examples of Odd Integers

Lastly, here are some examples of odd integers: -5, -3, -1, 1, 3, 5, 7, etc. All these, when divided by 2, would leave a remainder of 1 thus confirming that they are odd integers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Integers
Integers are whole numbers that can be either positive or negative, including zero. These numbers do not include fractions or decimals. Visualize them as points on a number line, where they extend infinitely in both directions. Examples of integers include: 3, -2, 0, and 7.

The term 鈥渋nteger鈥� originates from a Latin word meaning whole or complete, highlighting their nature as complete units without fractional parts. They are fundamental in mathematics as they serve as the building blocks for more complex number sets.

  • Positive integers: 1, 2, 3, etc.
  • Negative integers: -1, -2, -3, etc.
  • Zero: Neither positive nor negative, but it's an integer.
Exploring Number Sets
Number sets categorize numbers into groups based on their characteristics, making it easier to understand their properties. These sets include subsets like natural numbers, whole numbers, integers, rational numbers, and more. Each set shares particular properties based on how numbers within the set behave mathematically.

For example, integers are a subset of whole numbers that include positive numbers, negative numbers, and zero. They are part of a broader classification:
  • Natural Numbers: 1, 2, 3, etc. Known as counting numbers, not including zero.
  • Whole Numbers: 0, 1, 2, etc. Natural numbers plus zero.
  • Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Rational Numbers: Any number that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero.

Each of these number sets holds significance in various aspects of mathematical operations and problem-solving.
Remainders in Division
The remainder is a key concept in division when numbers are not perfectly divisible. It is the amount left over after dividing one integer by another. Understanding remainders can help us define certain types of numbers, such as odd and even integers.

When dividing by 2, the remainder can only be 0 or 1, which directly helps in identifying odd and even numbers:
  • Even integer: Leaves a remainder of 0 when divided by 2 (e.g., 4, 6, 8).
  • Odd integer: Leaves a remainder of 1 (e.g., 3, 5, 7).

In mathematical terms, an integer 'n' is odd if it can be written as \(n = 2k + 1\) where 'k' is another integer. This form guarantees a remainder of 1 when 'n' is divided by 2. Understanding these concepts is fundamental as it supports number theory, which plays a significant role in various mathematical disciplines.

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Most popular questions from this chapter

Let \(x=8, y=4,\) and \(z=2 .\) Write each phrase as an algebraic expression, and evaluate it. Assume that no denominators are \(0 .\) Find the numerical coefficients of each term.

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Write an algebraic expression to denote each quantity. Assume that no denominators are \(0 .\) Write an expression that represents the value of \(t\) T-bills, each worth \(\$ 9,987\).

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