3.2 The Integers
3 min read•june 18, 2024
are the building blocks of mathematics, encompassing whole numbers that can be positive, negative, or zero. They're represented on a , with positive numbers to the right of zero and negative numbers to the left.
Understanding integers is crucial for basic arithmetic operations. Addition, subtraction, multiplication, and division of integers follow specific rules based on their signs. These operations form the foundation for more complex mathematical concepts.
Understanding Integers
Integers on number lines
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- Integers are whole numbers that can be positive, negative, or zero
- Positive integers located to the right of zero on a number line (1, 2, 3)
- Negative integers located to the left of zero on a number line (-1, -2, -3)
- Zero is neither positive nor negative and is located at the center of the number line
Comparison of integer values
- Comparing integers
- Integer to the right on the number line is greater than the integer to the left
- Greater than symbol (>) indicates one integer is larger than another (5 > 2)
- Less than symbol (<) indicates one integer is smaller than another (-3 < 1)
- Equal to symbol (=) indicates two integers have the same value (-4 = -4)
- Ordering integers
- Arrange integers from left to right on the number line to order from least to greatest (-5, -2, 0, 3, 7)
- Arrange integers from right to left on the number line to order from greatest to least (8, 4, 1, -1, -6)
- of an integer is its distance from zero on the number line, regardless of its sign
Arithmetic operations with integers
- Addition
- Add integers with the same sign (both positive or both negative) by adding their absolute values and keeping the sign ,
- Add integers with different signs by subtracting the absolute values and keeping the sign of the integer with the larger absolute value ,
- Subtraction
- Subtract integers by adding the opposite (additive inverse) of the subtrahend (number being subtracted) to the minuend (number being subtracted from)
- Subtract integers by adding the opposite (additive inverse) of the subtrahend (number being subtracted) to the minuend (number being subtracted from)
- Multiplication
- Multiply integers by multiplying their absolute values and following the sign rules:
- Same signs (both positive or both negative) result in a positive product ,
- Different signs (one positive and one negative) result in a negative product ,
- Multiply integers by multiplying their absolute values and following the sign rules:
- Division
- Divide integers by dividing their absolute values and following the sign rules (same as multiplication):
- Same signs (both positive or both negative) result in a positive quotient ,
- Different signs (one positive and one negative) result in a negative quotient ,
- Divide integers by dividing their absolute values and following the sign rules (same as multiplication):
Properties of Integers
- are integers divisible by 2 without a remainder
- are integers not divisible by 2 without a remainder
- include all integers, as they can be expressed as a ratio of two integers
- represent repeated multiplication of a base number by itself, with negative exponents indicating reciprocals
Key Terms to Review (21)
Absolute value: Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of direction. This means that absolute value is always non-negative, and it transforms both positive and negative numbers into their positive equivalents. Understanding absolute value is crucial for working with integers, as it allows for comparisons between numbers without concern for their signs.
Composite numbers: Composite numbers are natural numbers greater than one that have more than two distinct positive divisors, meaning they can be divided evenly by 1, themselves, and at least one other number. This property distinguishes them from prime numbers, which have exactly two distinct positive divisors. Understanding composite numbers is essential in number theory, as they are integral to various mathematical concepts, including factorization and the identification of prime numbers.
Coprime: Coprime numbers, also known as relatively prime or mutually prime numbers, are two or more integers that share no common positive divisor other than 1. This concept highlights the relationship between numbers and their factors, emphasizing how some pairs of integers can be entirely independent in terms of their divisibility. Recognizing coprime pairs is crucial in number theory and helps in understanding the properties of prime and composite numbers as well as the behavior of integers.
Diophantine equation: A Diophantine equation is a polynomial equation that allows for integer solutions only, often taking the form $ax + by = c$ where $a$, $b$, and $c$ are integers, and $x$ and $y$ are the unknowns that also need to be integers. This type of equation is named after the ancient Greek mathematician Diophantus, who studied such equations in depth. The solutions to these equations are not always guaranteed, and finding them can involve techniques from number theory.
Divisibility: Divisibility is a mathematical concept that determines whether one integer can be evenly divided by another integer without leaving a remainder. This concept plays a critical role in identifying prime and composite numbers, as a prime number has only two distinct positive divisors: 1 and itself, while composite numbers have more than two divisors. Understanding divisibility helps in performing operations involving integers and lays the groundwork for various mathematical theories and applications.
Euclid: Euclid was an ancient Greek mathematician often referred to as the 'Father of Geometry' for his influential work, 'Elements.' This comprehensive compilation laid the groundwork for modern mathematics by systematically presenting the principles of geometry and number theory, connecting various mathematical concepts through definitions, postulates, and proofs.
Euclidean algorithm: The Euclidean algorithm is a method for finding the greatest common divisor (GCD) of two integers, which is the largest positive integer that divides both numbers without leaving a remainder. This algorithm is essential in number theory, as it provides a systematic way to compute the GCD and has implications for various mathematical concepts such as fractions, modular arithmetic, and divisibility.
Even Numbers: Even numbers are integers that can be exactly divided by 2, meaning they have no remainder when divided. These numbers include both positive and negative values, as well as zero, forming an essential subset of the integers. Even numbers are crucial in understanding patterns in mathematics, such as divisibility and parity, and they play a significant role in various mathematical operations and properties.
Fermat: Fermat refers to Pierre de Fermat, a French lawyer and mathematician who is best known for his contributions to number theory, particularly regarding the properties of integers. His famous last theorem, which states that there are no three positive integers a, b, and c that satisfy the equation $$a^n + b^n = c^n$$ for any integer value of n greater than 2, sparked interest and research in the field for centuries. Fermat's work laid foundational concepts in the study of integers and modular arithmetic.
Fundamental Theorem of Arithmetic: The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed uniquely as a product of prime numbers, up to the order of the factors. This theorem establishes the importance of prime numbers as the building blocks of all integers, highlighting their role in number theory and reinforcing the distinction between prime and composite numbers.
GCD: The GCD, or Greatest Common Divisor, is the largest positive integer that divides two or more integers without leaving a remainder. It is a fundamental concept in number theory and helps in understanding the relationships between numbers, particularly in the context of prime and composite numbers as well as integers. The GCD can reveal whether two numbers share any common factors, making it essential for simplifying fractions and solving problems involving divisibility.
Integer exponents: Integer exponents are a way to express repeated multiplication of a number by itself, where the exponent is a whole number that can be positive, negative, or zero. This concept is key in simplifying expressions and solving equations, as it helps in handling powers efficiently. Understanding integer exponents allows for the application of rules such as the product of powers and the power of a power, which are essential in various mathematical operations.
Integers: Integers are whole numbers that can be positive, negative, or zero, encompassing all the numbers without fractional or decimal components. They play a crucial role in various mathematical concepts, including arithmetic operations, number theory, and the structure of real numbers. Integers are fundamental for understanding prime and composite numbers, as these classifications are specific to the set of positive integers.
Mod: The term 'mod' refers to the modulo operation, which determines the remainder when one integer is divided by another. It's a way to express relationships between numbers in a cyclical manner, often useful in various mathematical fields, especially in number theory and applications involving periodic phenomena, like clock arithmetic. The result of the operation is always non-negative and is less than the divisor.
Modular arithmetic: Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after reaching a certain value, known as the modulus. This type of arithmetic is used to simplify calculations by focusing on remainders rather than complete values. It's especially useful in various mathematical applications like cryptography, computer science, and number theory.
Net worth: Net worth is the difference between an individual's or entity's assets and liabilities. It represents the value of everything owned minus the debts owed.
Number line: A number line is a visual representation of numbers arranged in a straight horizontal line, where each point corresponds to a real number. It serves as a tool to understand numerical relationships and operations, providing a clear way to visualize integers, rational numbers, and real numbers. The concept of the number line is crucial for solving equations and inequalities, helping to illustrate how different types of numbers are related and how they can be compared or combined.
Odd numbers: Odd numbers are integers that cannot be exactly divided by 2, resulting in a remainder of 1 when divided. They are characterized by their inability to form pairs, meaning they are not evenly divisible by two. This unique property makes odd numbers important in various mathematical concepts, including number theory and arithmetic sequences.
Prime factorization: Prime factorization is the process of breaking down a composite number into the product of its prime factors. This method reveals the fundamental building blocks of a number, demonstrating how it can be expressed as a multiplication of primes. Understanding prime factorization is essential for various mathematical concepts, including finding the greatest common divisor and least common multiple.
Prime numbers: Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. They are the building blocks of the integers, as every integer greater than 1 can be expressed as a product of prime numbers. Understanding prime numbers is crucial for various mathematical concepts, including factorization, number theory, and cryptography.
Rational Numbers: Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. This definition connects them to whole numbers, integers, and other types of numbers, making them a fundamental part of the number system. They can be positive, negative, or zero and can also be represented as terminating or repeating decimals.