1. 1 Topic 1.4.1 Mathematical Proofs

2. 2 Topic 1.4.1 Mathematical Proofs California Standards: 24.2 Students identify the hypothesis and conclusion in logical deduction. 25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step. What it means for you: You’ll justify each step of mathematical proofs, and you’ll learn about the hypothesis and conclusion of “if... then” statements. Key words: justify hypothesis conclusion proof

3. 3 Topic 1.4.1 Mathematical Proofs A lot of Algebra I asks you to give formal proofs for stuff that you covered in earlier grades. You’re sometimes asked to state exactly which property you’re using for every step of a math problem.

4. 4 Topic 1.4.1 You Must Justify Each Step of a Mathematical Proof Mathematical Proofs A mathematical proof is a logical argument. When you write a mathematical proof, you have to justify each step in a logical way. In Algebra I, you do this using the axioms covered earlier in this chapter. You’ve seen lots of proofs already in this chapter — although some of them weren’t described as proofs at the time. Solving an equation to find the value of a variable is a form of mathematical proof.

5. 5 Topic 1.4.1 Mathematical Proofs The next Example shows a mathematical proof. It is written in two columns — with each step of the logical argument written on the left, and the justification for it written on the right.

6. 6 Topic 1.4.1 Example 1 Solution follows… Mathematical Proofs If 6 x + 4 = 22, what is the value of x ? Solution 6 x + 4 = 22 (6 x + 4) – 4 = 22 – 4 (6 x + 4) + (–4) = 22 – 4 6 x + (4 + (–4)) = 22 – 4 6 x + (4 + (–4)) = 18 6 x + 0 = 18 6 x = 18 Given equation Subtraction property of equality Definition of subtraction Associative property of addition Subtracting Inverse property of addition Identity property of addition Solution continues…

7. 7 Topic 1.4.1 Example 1 Mathematical Proofs … 6 x = 18 Solution (continued) 1 x = 3 x = 3 Inverse property of multiplication Identity property of multiplication Multiplication property of equality × (6 x ) = × 18 1 6 1 6 Associative property of multiplication × 6 x = × 18 1 6 1 6 Definition of division × 6 x = 1 6 18 6 Dividing × 6 x = 3 1 6

8. 8 Topic 1.4.1 Guided Practice Solution follows… Mathematical Proofs Complete these statements: 1. A mathematical proof is called a....................... because you have to................. each step in a logical way using mathematical................. 2. Mathematical proofs can be written in two columns, with the ….................... on the left and the................... on the right. logical argument justify axioms justifications logical argument

9. 9 Topic 1.4.1 Proofs Can Often be Shortened by Combining Steps Mathematical Proofs Proofs can very often be written in the kind of two-column format used in the last example. The next statement in your argument goes on the left, and the justification for it goes on the right. Usually the justification will be something from earlier in this chapter. However, it’s not likely that you’d often need to include every single possible stage in a proof. Usually you’d solve an equation in a few lines, as shown in Example 2.

10. 10 Topic 1.4.1 Example 2 Solution follows… Mathematical Proofs If 6 x + 4 = 22, what is the value of x ? Solution 6 x + 4 = 22 6 x = 18 x = 3

11. 11 Topic 1.4.1 Mathematical Proofs Usually it’s quicker (and a much better idea) to solve an equation the short way, like in Example 2. But you must be able to do it the long way if you need to, justifying each step using the real number axioms.

12. 12 Topic 1.4.1 “If…, Then…” Gives a Hypothesis and a Conclusion Mathematical Proofs Mathematical statements can often be written in the form: “If..., then...” For example, when you solve an equation like the one in Example 2, what you are really saying is: “If 6 x + 4 = 22, then the value of x is 3.” A sentence like this can be broken down into two basic parts — a hypothesis and a conclusion. The hypothesis is the part of the sentence that follows “if” — here, it is 6 x + 4 = 22. The conclusion is the part of the sentence that follows “then” — here, it is x = 3.

13. 13 Topic 1.4.1 Mathematical Proofs IF hypothesis, THEN conclusion. This doesn’t just apply to mathematical statements — it’s true for non-mathematical “If..., then...” sentences as well. If an animal is an insect, then it has six legs. If you are in California, then you are in the United States. For example:

14. 14 Topic 1.4.1 Mathematical Proofs However, the conclusion has to be a logical consequence of the hypothesis. Using the insect example, this just means that if it is an insect, then it will have six legs. Now, both the hypothesis and the conclusion can be either true or false. For example, an animal may or may not be an insect, and it may or may not have six legs.

15. 15 Topic 1.4.1 Mathematical Proofs Once you’ve figured out a hypothesis and a conclusion, you can apply the following logical rules: If the hypothesis is true, then the conclusion will also be true. If the conclusion is false, then the hypothesis will also be false. So if an animal doesn’t have six legs, then it isn’t an insect. If you aren’t in the United States, then you’re not in California. And if x is not 3, then 6 x + 4  22.

16. 16 Topic 1.4.1 Independent Practice Solution follows… Mathematical Proofs Rewrite the following in “If..., then...” format. 1. 4 x = 12 means that x = 3. 2. x + y = 1 means that x = 1 – y. 3. b + 4 = 17 – y means that b = 13 – y. If 4 x = 12, then x = 3. If x + y = 1, then x = 1 – y. If b + 4 = 17 – y, then b = 13 – y.

17. 17 Topic 1.4.1 Independent Practice Solution follows… Mathematical Proofs Identify the hypothesis and conclusion in the following statements. 4. If 5 y = 30, then y = 6. 5. If x 2 + y 2 = 16, then x 2 = 16 – y 2. 6. If d – 12 = 23 z, then d = 23 z + 12. 7. An animal has four legs if it is a dog. Hypothesis: 5 y = 30 Conclusion: y = 6 Hypothesis: x 2 + y 2 = 16 Conclusion: x 2 = 16 – y 2 Hypothesis: d – 12 = 23 z Conclusion: d = 23 z + 12 Hypothesis: An animal is a dog. Conclusion: It has four legs.

18. 18 Topic 1.4.1 Independent Practice Solution follows… Mathematical Proofs 8. Complete this proof by adding the missing justifications. x – 7 = 17Given equation ( x – 7) + 7 = 17 + 7........................................... [ x + (–7)] + 7 = 17 + 7Definition of subtraction [ x + (–7)] + 7 = 24Adding x + [(–7) + 7] = 24........................................... x + 0 = 24Inverse property of addition x = 24........................................... Addition property of equality Associative property of addition Identity property of addition

19. 19 Topic 1.4.1 Round Up Mathematical Proofs The important thing with mathematical proofs is to take each line of the math problem step by step. If you’re asked to justify your steps, make sure that you state exactly which property you’re using.