Fill in the Blanks of the Following Proof by Contradiction $7 + 4 \sqrt { 2 }$

question 17

Essay

Fill in the blanks of the following proof by contradiction that $7 + 4 \sqrt { 2 }$ is an irrational number. (You may use the fact that $\sqrt { 2 }$ is irrational.)
Proof: Suppose not. That is, suppose that $7 + 4 \sqrt { 2 }$ is (i). By definition of rational, $7 + 4 \sqrt { 2 } =$ $\frac { a } { b }$ , where (ii). Multiplying both sides by $b$ gives
$$7 b + 4 b \sqrt { 2 } = a ,$$
so if we subtract $7 b$ from both sides we have
$$4 b \sqrt { 2 } = \underline { \text { (iii) } } .$$
Dividing both sides by $4 b$ gives
$$\sqrt { 2 } = \text { (iv). }$$
But then $\sqrt { 2 }$ would be a rational number because (v). This contradicts our knowledge that $\sqrt { 2 }$ is irrational. Hence (vi).

Understand how migration affects labor supply curves in origin and destination countries.

Analyze the effects of wage changes on labor market equilibrium, including supply, demand, and wages.

Comprehend the impact of leisure opportunity cost on work decisions and labor supply.

Examine how external changes, such as immigration or industry attractiveness, influence labor markets.

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