# The integer part is (? distributive ?)

In summary: So if [x] = n then x = n + f. If [x - 1] = [x] - 1, then x - 1 = n - 1 + f. So [x - 1] = n - 1, and the result follows.In summary, the floor of a real number k, denoted as [k], is the greatest integer that is less than or equal to k. It can also be represented as k rounded down to the nearest whole number. The formula [a - b] = [a] - [b] holds true for any positive real number a and b, when b is either 0 or 1. This can be proven by writing
the integer part is ... (?? distributive ??)

## Homework Statement

Define the floor of a real number k where [k] is the least smallest integer from k.

I want to show that [a - b] = [a] -

n/a

## The Attempt at a Solution

[1.2 - 5.7] = [-3.8] = -4
[1.2] - [5.7] = 1 - 5 = -4

I am not sure how to go about generalizing the observation above to all numbers.

You don't:
[5.2 - 2.3] = [2.9] = 2
[5.2] - [2.3] = [5] - [2] = 3

Sorry

Yes, thank you CompuChip. However I stated the problem wrong. What I am thinking is this:

Let a > 0. Then
[a - 0] = [a] - 0 and
[a - 1] = [a] - 1.

I only care about the values being subtracted from a of 0 and 1, nothing else. Then if b = 0 or 1, [a-b] = [a] - b. This should hold, correct? Demonstrating it works for 0 is easy. I don't know how to show it for 1, but here is my attempt:

a - 1 < a
Suppose a is an integer. Then [a - 1] = a - 1 = [a] - 1.
Suppose a is not an integer. I am not sure. It seems to work fine with all the tests I can give (a>0). I don't know how to write this.

Let x be any (positive, although it probably works for any) number. Then write x as n + f, with n an integer and f the fractional part (0 <= f < 1).

CompuChip said:
Let x be any (positive, although it probably works for any) number. Then write x as n + f, with n an integer and f the fractional part (0 <= f < 1).

Yes, thank you. Work backwards, of course.

## What does "the integer part" refer to in this context?

The integer part refers to the whole number portion of a decimal or fractional number. For example, in the number 4.32, the integer part is 4.

## What does the term "distributive" mean in this context?

In mathematics, the distributive property refers to the ability to break down an expression into smaller parts and then perform operations on each part separately. In the context of the integer part, this means that the distributive property can be used to simplify expressions involving integers.

## How is the distributive property used with the integer part?

The distributive property can be used with the integer part to simplify expressions involving integers. For example, if we have the expression 3(x + 2), we can use the distributive property to rewrite it as 3x + 6, where the integer part (3) is distributed to each term inside the parentheses.

## Can the distributive property be used with all types of numbers?

Yes, the distributive property can be used with all types of numbers, including integers. It is a fundamental property of mathematics that applies to all numbers and operations.

## Why is the distributive property important in mathematics?

The distributive property is important in mathematics because it allows us to simplify and solve complex expressions and equations. It is a key concept in algebra and is used in many other areas of mathematics, such as calculus and statistics.

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