proof - an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion
A 13 ft. ladder is leaning against a building 12 ft. up from the ground. How far is the base of theA 13 ft. ladder is leaning against a building 12 ft. up from the ground. How far is the base of the ladder from the building?
This is a classic 5-12-13 pythagorean triple, where the hypotenuse is 13, and the 2 sides are 5 and 12. The building and the ground form a right triangle.
[URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=12&hypinput=13&pl=Solve+Missing+Side']You can see the proof here[/URL]...
Additive Identity PropertyFree Additive Identity Property Calculator - Displays the line by line proof for the additive identity property
Numerical Properties
C is the midpoint of BD then BC congruent CDC is the midpoint of BD then BC congruent CD
[URL='https://www.mathcelebrity.com/proofs.php?num=cisthemidpointofbd&pl=Prove']True using this proof[/URL]
Consecutive Interior AnglesFree Consecutive Interior Angles Calculator - Shows you a proof of consecutive interior angles using parallel lines and a transversal
DeMorgans LawsFree DeMorgans Laws Calculator - Demonstrates DeMorgans Laws including the proof
Let n be an integer. If n^2 is odd, then n is oddLet n be an integer. If n^2 is odd, then n is odd
Proof by contraposition:
Suppose that n is even. Then we can write n = 2k
n^2 = (2k)^2 = 4k^2 = 2(2k) so it is even
[I]So an odd number can't be the square of an even number. So if an odd number is a square it must be the square of an odd number.[/I]
Let x be an integer. If x is odd, then x^2 is oddLet x be an integer. If x is odd, then x^2 is odd
Proof: Let x be an odd number. This means that x = 2n + 1 where n is an integer.
[U]Squaring x, we get:[/U]
x^2 = (2n + 1)^2 = (2n + 1)(2n + 1)
x^2 = 4n^2 + 4n + 1
x^2 = 2(2n^2 + 2n) + 1
2(2n^2 + 2n) is an even number since 2 multiplied by any integer is even
So adding 1 is an odd number
[MEDIA=youtube]GlzV80M33x0[/MEDIA]
ProofsFree Proofs Calculator - Various Proofs in Algebra
Prove 0! = 1[URL='https://www.mathcelebrity.com/proofs.php?num=prove0%21%3D1&pl=Prove']Prove 0! = 1[/URL]
Let n be a whole number, where n! represents:
The product of n and all integers below it through 1.
The factorial formula for n is
n! = n · (n - 1) · (n - 2) · ... · 3 · 2 · 1
Written in partially expanded form, n! is:
n! = n · (n - 1)!
[SIZE=5][B]Substitute n = 1 into this expression:[/B][/SIZE]
n! = n · (n - 1)!
1! = 1 · (1 - 1)!
1! = 1 · (0)!
For the expression to be true, 0! [U]must[/U] equal 1.
Otherwise, 1! ≠ 1 which contradicts the equation above
[MEDIA=youtube]wDgRgfj1cIs[/MEDIA]
Prove sqrt(2) is irrationalUse proof by contradiction. Assume sqrt(2) is rational.
This means that sqrt(2) = p/q for some integers p and q, with q <>0.
We assume p and q are in lowest terms.
Square both side and we get:
2 = p^2/q^2
p^2 = 2q^2
This means p^2 must be an even number which means p is also even since the square of an odd number is odd.
So we have p = 2k for some integer k. From this, it follows that:
2q^2 = p^2 = (2k)^2 = 4k^2
2q^2 = 4k^2
q^2 = 2k^2
q^2 is also even, therefore q must be even.
So both p and q are even.
This contradicts are assumption that p and q were in lowest terms.
So sqrt(2) [B]cannot be rational.
[MEDIA=youtube]tXoo9-8Ewq8[/MEDIA][/B]
Pythagorean TheoremFree Pythagorean Theorem Calculator - Figures out based on user entry the missing side or missing hypotenuse of a right triangle. In addition, the calculator shows the proof of the Pythagorean Theorem and then determines by numerical evaluation if the 2 sides and hypotenuse you entered are a right triangle using the Pythagorean Theorem
Pythagorean Theorem Trig ProofsFree Pythagorean Theorem Trig Proofs Calculator - Shows the proof of 3 pythagorean theorem related identities using the angle θ:
Sin2(θ) + Cos2(θ) = 1
Tan2(θ) + 1 = Sec2(θ)
Sin(θ)/Cos(θ) = Tan(θ)
Rule of 72Free Rule of 72 Calculator - Calculates how long it would take money to double (doubling time) using the rule of 72 interest approximation as well as showing the mathematical proof of the Rule of 72.
Tamara can proofread 16 pages in 8 minutes. How many minutes will it take her to proofread 108 pagesTamara can proofread 16 pages in 8 minutes. How many minutes will it take her to proofread 108 pages
Set up a proportion of pages to minutes:
16 pages/8 minutes = 108 pages / p minutes
We want to solve for p.
Type [I][URL='https://www.mathcelebrity.com/prop.php?num1=16&num2=108&den1=8&den2=p&propsign=%3D&pl=Calculate+missing+proportion+value']16/8 = 108/p[/URL][/I] into the search engine.
We get p = [B]54 minutes[/B]
The difference between the squares of two consecutive numbers is 141. Find the numbersThe difference between the squares of two consecutive numbers is 141. Find the numbers
Take two consecutive numbers:
n- 1 and n
Given a difference (d) between the squares of two consecutive numbers, the shortcut for this is:
2n - 1 = d
Proof of this:
n^2- (n - 1)^2 = d
n^2 - (n^2 - 2n + 1) = d
n^2 - n^2 + 2n - 1 = d
2n - 1 = d
Given d = 141, we have
2n - 1 = 141
Add 1 to each side:
2n = 142
Divide each side by 2:
2n/2 = 142/2
n = [B]71[/B]
Therefore, n - 1 = [B]70
Our two consecutive numbers are (70, 71)[/B]
Check your work
70^2 = 4900
71^2 = 5041
Difference = 5041 - 4900
Difference = 141
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Two Column ProofFree Two Column Proof Calculator - Shows you the details behind a two column proof including the five parts and examples