Real numbers: Basic Operations
A real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers (which can be expressed as a ratio of integers), such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (which can't be expressed as a ratio of integers). Real numbers can be thought of as points on an infinitely long line called the number line (real line), where the points corresponding to integers are equally spaced as shown in the figure below.
Real numbers
Real numbers can be thought of as points on an infinitely long number line.
The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division. Arithmetic operations are performed according to a specific hierarchy or order, not from left to right.
Addition and Subtraction
Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers into a single number. Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series. Addition is commutative and associative, so the order in which the terms are added does not affect their sum.
For example, 3+(2+5)=3+7=10 is the same as (3+2)+5=5+5=10. Even though the groupings of the numbers are different, they add to the same number because addition is associative.
Addition is also commutative, meaning that the order does not matter. So 2+3+5 is the same as 5+2+3.
The identity element of addition is 0; that is, adding zero to any number yields that same number.
Subtraction is the reverse of addition; it finds the difference between two numbers. As such, taking a number x, adding another number, b, to it and subsequently subtracting b from it affords the same number x. That is, x+b−b=x. Subtraction is neither commutative nor associative.
For example, 3−(2+5)=3−7=−4 is different than (3−2)+5=1+5=6 because subtraction is not associative.
2−3+5 is also different than 3−2+5, with the first expression equaling 4 and the second equaling 6. Therefore, subtraction is not commutative.
Multiplication and Division
Multiplication also combines two numbers into a single number, the product. Multiplication is best viewed as a simplification of many additions. For example, the product of x and y is the sum of x written out y times.
Multiplication is commutative and associative, and its identity is 1. That is, multiplying any number by 1 yields that same number. Division is the inverse of multiplication. Thus, taking a number x and multiplying it by b and then dividing it by b results in the same number x. Like subtraction, division is neither commutative nor associative.
A real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers (which can be expressed as a ratio of integers), such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (which can't be expressed as a ratio of integers). Real numbers can be thought of as points on an infinitely long line called the number line (real line), where the points corresponding to integers are equally spaced as shown in the figure below.
Real numbers
Real numbers can be thought of as points on an infinitely long number line.
The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division. Arithmetic operations are performed according to a specific hierarchy or order, not from left to right.
Addition and Subtraction
Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers into a single number. Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series. Addition is commutative and associative, so the order in which the terms are added does not affect their sum.
For example, 3+(2+5)=3+7=10 is the same as (3+2)+5=5+5=10. Even though the groupings of the numbers are different, they add to the same number because addition is associative.
Addition is also commutative, meaning that the order does not matter. So 2+3+5 is the same as 5+2+3.
The identity element of addition is 0; that is, adding zero to any number yields that same number.
Subtraction is the reverse of addition; it finds the difference between two numbers. As such, taking a number x, adding another number, b, to it and subsequently subtracting b from it affords the same number x. That is, x+b−b=x. Subtraction is neither commutative nor associative.
For example, 3−(2+5)=3−7=−4 is different than (3−2)+5=1+5=6 because subtraction is not associative.
2−3+5 is also different than 3−2+5, with the first expression equaling 4 and the second equaling 6. Therefore, subtraction is not commutative.
Multiplication and Division
Multiplication also combines two numbers into a single number, the product. Multiplication is best viewed as a simplification of many additions. For example, the product of x and y is the sum of x written out y times.
Multiplication is commutative and associative, and its identity is 1. That is, multiplying any number by 1 yields that same number. Division is the inverse of multiplication. Thus, taking a number x and multiplying it by b and then dividing it by b results in the same number x. Like subtraction, division is neither commutative nor associative.