Proof in Algebra 3-8
Some of the properties discussed in the previous chapters are statements we assume to be true. Others are called theorems. A theorem is a statement that is shown to be true using a logically developed argument. Logical reasoning that uses given facts, definitions, properties, and other already proven theorems to show that a particular theorem is true is called a proof. Proofs are used extensively in Algebra as well as Geometry.
Prove: For all numbers a and b, (a + b) – b = a
Many times, only the KEY reasons are states—the substitute principle and the properties of equality are usually not stated. So, the above prove could be shortened to 4 steps:
Prove: For all real numbers a and b, such that a≠ 0 and b ≠0
1/ab = 1/a ⋅1/b
Since 1/ab is the unique reciprocal of ab, you can prove that
1/ab = 1/a⋅ 1/b by showing that the product of ab and 1/a ⋅ 1/b is 1
Once a theorem has been proved, you can use it as a reason in other proofs. Check the Chapter summary on page 88 of our textbook for the listing of properties and theorems that you can use as reasons in your proofs for our homework.
