The Real Numbers
Real numbers are closed under addition, subtraction, multiplication, and division. This means that if you take two real numbers and perform one of these operations on them, the result will also be a real number. For example, 2 + 5 = 7, which is a real number. Similarly, 5 – 7 = -2, which is also a real number.
The Set of Real Numbers
All real numbers are either rational or irrational. A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q is not equal to 0. An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q.
The Properties of Real Numbers
There are many different properties of real numbers, which include the closure properties, commutative properties, associative properties, distributive property, identity property, inverse property, and powers property. The closure properties state that if two numbers are added or multiplied together, the result will always be a real number. For example, 2 5 72 5 7. The commutative properties state that the order of addition or multiplication does not matter, for example 2 5 7 5 2. The associative properties state that when adding or multiplying more than two numbers together, the grouping of the numbers does not matter as long as the order within each group stays the same. For example, (2 5) 7 6 (5 7). The distributive property states that when multiplying a number by a sum of two other numbers, you can multiply the first number by each of the other two numbers and then add these products together to get the same result. For example, 3(4 5) 3(4) 3(5) 12 15. The identity property states that when you add or multiply any number by 0, the result will always be equal to 0. For example 0 2 0 or 3 0 0. The inverse property states that every number has an inverse or opposite that when added to the original number will equal 0. For example 2 5 7 because 7 2 5 0. And finally, the powers property states that when dividing like terms with exponents, you can divide the coefficients and subtract the exponents to get your answer. For example 23 21 23 21=23-1=2 .
The Equation 2 5 72 5 7
The equation 2 5 72 5 7 is an example of the commutative property of real numbers. This property states that when two numbers are added together, the order in which they are added does not affect the sum. In other words, the equation 2 5 72 5 7 is equivalent to the equation 5 2 7 5 7 2.
The Left Side of the Equation
The equation 2 5 72 5 7 illustrates the property of real numbers that says the left side of the equation is always equal to the right side of the equation. This is true for all real numbers, no matter what their value is.
The Right Side of the Equation
When solving equations, we are looking for the value(s) of the variable(s) that make the equation true. In other words, we are looking for what goes on the left side of the equation when the equation is true. For example, in the equation 2 + 5 = 7, we are looking for what number we can put in place of the + sign to make the equation true. In this case, the answer is 2. So, 2 + 5 does indeed equal 7.
The Solution to the Equation
The answer to the equation 2 5 72 5 7 is 6. This equation illustrates the property of real numbers known as the distributive property.