Describe the relationship between integers and whole numbers. Are all integers also whole numbers? Are all whole numbers also integers? Give examples to support your answer.(4 poi
All integers are also whole numbers, but not all whole numbers are integers.
Integers are the set of counting numbers, their negatives, and zero. Whole numbers are the set of counting numbers and zero.
For example, the number 3 is an integer because it is a counting number. It is also a whole number because it is a counting number and zero.
However, the number -2 is an integer because it is the negative of a counting number. It is not a whole number because it does not include zero.
Therefore, every integer is also a whole number since it includes zero, but not every whole number is an integer since it may not include negative numbers.
The diagonal of a square equals 2–√
times the side length. Using 1.4 for the square root of 2, what is the approximate length in feet from one corner to an opposite corner of a square park that is 70 feet on a side?(1 point)
ft.
The length of the diagonal of a square is given by the formula diagonal = side length * (2-√2).
Given that the square root of 2 is approximately 1.4, we can substitute this value into the formula:
diagonal = 70 * (2-1.4) = 70 * 0.6 = 42
Therefore, the approximate length from one corner to an opposite corner of the square park is 42 feet.
Convert 0.09¯¯¯¯¯ to a rational number in simplest form.(1 point)
To convert the repeating decimal 0.09¯¯¯¯¯ to a rational number in simplest form, we need to set up an equation.
Let x = 0.09¯¯¯¯¯
We can multiply both sides of the equation by 100 to remove the repeating decimal:
100x = 9.999¯¯¯¯¯
Now, we subtract the original equation from the new equation:
100x - x = 9.999¯¯¯¯¯ - 0.09¯¯¯¯¯
99x = 9.909¯¯¯¯¯
We subtract the original equation from the new equation:
99x = 9.909¯¯¯¯¯ - 0.09¯¯¯¯¯
99x = 9.819¯¯¯¯¯
Now we can write the repeating decimal in fraction form:
$99 x$ = $9.819819819819...$
To convert the repeating decimal to a fraction, we assign a variable to the repeating part.
Let y = 0.819819819819...
Now, we can subtract y from the left side of the equation to obtain:
99y = 9.819819819819... - 0.819819819819...
Simplifying the equation:
99y = 9
Now, we divide both sides of the equation by 99:
y = 9/99
Since y represents the repeating part of the original decimal, we need to add it to the non-repeating part, 0.09.
0.09 + y = 0.09 + 9/99
Combining like terms:
0.09 + 9/99 = 0.09 + 1/11 = 0.09 + 0.090909090909...
Now, we have a repeating fraction. To simplify it, we can again assign a variable to the repeating part:
Let z = 0.090909090909...
Subtracting z from the original equation:
z = 0.090909090909... - 0.09
Simplifying the equation:
z = 1/11
Now, we can add the non-repeating part, 0.09, to the repeating fraction:
0.09 + z = 0.09 + 1/11
Combining like terms:
0.09 + 1/11 = (9/100) + (1/11)
Adding the fractions:
(9/100) + (1/11) = 19/100
Therefore, the rational number equivalent to the repeating decimal 0.09¯¯¯¯¯ in simplest form is 19/100.
The relationship between integers and whole numbers is that every integer is a whole number, but not every whole number is an integer.
Integers include positive, negative, and zero whole numbers. They can be represented by the set of numbers {..., -3, -2, -1, 0, 1, 2, 3, ...}.
On the other hand, whole numbers consist of only positive whole numbers and zero. They can be represented by the set of numbers {0, 1, 2, 3, ...}. Whole numbers do not include negative numbers.
To further illustrate this relationship, consider the following examples:
1. The number -5 is an integer because it belongs to the set of integers {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, ...}. It is also a whole number since it is an integer without any fractional parts.
2. The number 4 is an integer because it belongs to the set of integers {..., -3, -2, -1, 0, 1, 2, 3, 4, ...}. It is also a whole number since it is an integer without any fractional parts.
3. The number 0 is an integer because it belongs to the set of integers {..., -2, -1, 0, 1, 2, 3, ...}. It is also a whole number since it is an integer without any fractional parts.
4. The number 2.5 is not an integer because it is a non-whole number. It has a fractional part, making it a real number, but not an integer or a whole number.
In summary, while all integers are also whole numbers, not all whole numbers are integers. Integers include both positive and negative whole numbers, while whole numbers only consist of positive whole numbers and zero.
Integers and whole numbers are related to each other in the sense that integers are a superset of whole numbers. In other words, all whole numbers can be considered integers, but not all integers are whole numbers.
Whole numbers include all positive numbers starting from zero, while integers include both positive and negative whole numbers, along with zero.
For example, the whole numbers 0, 1, 2, 3, 4, 5, and so on are also integers. However, the negative whole numbers such as -1, -2, -3, and so on are included in the set of integers but not in the set of whole numbers. Additionally, zero is both a whole number and an integer.
So, to summarize:
- All whole numbers are integers.
- Not all integers are whole numbers.
Here are some examples to illustrate this relationship:
- 0 is both a whole number and an integer.
- 5 is a whole number and also an integer.
- -3 is an integer but not a whole number.