Integers are a set of positive and negative whole numbers, including zero. They are the basic building blocks of mathematics and are used in a wide range of applications, from everyday calculations to advanced scientific and mathematical models.
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Integers can be positive, negative, or zero, and they have no fractional part.
The set of integers is denoted by the symbol $\mathbb{Z}$, which stands for the German word 'Zahlen' meaning 'numbers'.
Integers follow the same basic arithmetic operations (addition, subtraction, multiplication, and division) as natural numbers, but with the inclusion of negative numbers.
The absolute value of an integer is the distance of the number from zero on the number line, regardless of its sign.
Integers are closed under the operations of addition, subtraction, and multiplication, but not division (except for division by a non-zero integer).
Review Questions
Explain the properties of integers that distinguish them from other number systems, such as rational numbers or real numbers.
Integers are a unique set of numbers that have several distinguishing properties. Unlike rational numbers, which can be expressed as fractions, integers are whole numbers that have no fractional part. Additionally, unlike real numbers, which include irrational numbers like $\pi$ and $\sqrt{2}$, integers are discrete and can be ordered on the number line. This means that between any two integers, there are no other integers, whereas between any two real numbers, there are infinitely many other real numbers. These properties make integers the fundamental building blocks of mathematics and arithmetic.
Describe the relationship between integers and the operations of addition and subtraction. How do these operations affect the sign and magnitude of integers?
Integers are closed under the operations of addition and subtraction, meaning that the sum or difference of two integers will always result in another integer. When adding two integers, the signs of the numbers determine the sign of the result: adding a positive and a negative integer results in a number with the sign of the larger absolute value, while adding two integers with the same sign results in a number with that same sign. Subtraction of integers is related to addition, as subtracting a number is the same as adding its additive inverse. The absolute value of the result of an addition or subtraction operation is the distance between the two integers on the number line.
Analyze the role of integers in the development of mathematical concepts, such as the number line, absolute value, and the additive inverse. How do these concepts contribute to the understanding and application of integers?
Integers are fundamental to the development of many mathematical concepts that are essential for understanding and working with numbers. The number line, for example, is a visual representation of the set of integers, with positive integers to the right of zero and negative integers to the left. This allows for the ordering and comparison of integers. The concept of absolute value, which represents the distance of a number from zero on the number line, is also closely tied to the properties of integers. Additionally, the idea of the additive inverse, where every integer has a corresponding number that, when added, results in zero, is a key property that enables the subtraction of integers and the solving of equations. These concepts, built upon the foundation of integers, are crucial for the further development of mathematical reasoning and problem-solving skills.