The World of Numbers | EOT

Question 9

Let $a$ and $b$ be two non-zero rational numbers such that $a + \dfrac{1}{b} = 0$ . Without assigning any numerical values, determine whether $ab$ is positive or negative. Justify your answer.

Solution

Understand the Question

We are given two non-zero rational numbers $a$ and $b$ satisfying the equation $a + \dfrac{1}{b} = 0$ .
To find whether the product $ab$ is positive or negative without using numerical examples, we algebraically rearrange the equation to solve directly for $ab$ .
We then determine the sign of the resulting value.

Step 1 · Find the Product $ab$

Given $a + \dfrac{1}{b} = 0$

Rearranging the equation to isolate $a$ $a = -\dfrac{1}{b}$

Multiplying both sides by $b$ (since $b \neq 0$ ) $ab = -1$

Since $-1 < 0$ , the product $ab$ is always negative.

Answer

Negative ( $ab = -1$ )

Common Mistakes

Assigning Numerical Values: Substituting specific numbers instead of using algebraic proof, which violates the requirement "without assigning any numerical values".
Sign Errors in Transposition: Mistakenly writing $a = \dfrac{1}{b}$ instead of $a = -\dfrac{1}{b}$ , which incorrectly gives $ab = 1$ (positive).

More questions in EOT

Convert the following rational numbers in the form of a terminating decimal or non-terminating and repeating decimal, whichever the case may be, by the process of long division:

(i) $\dfrac{3}{50}$

(ii) $\dfrac{2}{9}$

Prove that $\sqrt{5}$ is an irrational number.

Convert the following decimal numbers in the form of $\dfrac{p}{q}$ .

(i) $12.6$

(ii) $0.0120$

(iii) $3.05\overline{2}$

(iv) $1.2\overline{35}$

(v) $0.\overline{23}$

(vi) $2.0\overline{5}$

(vii) $2.12\overline{5}$

(viii) $3.12\overline{5}$

(ix) $2.\overline{1625}$

Locate the following rational numbers on the number line.

(i) $0.532$

(ii) $1.1\bar{5}$

Find 6 rational numbers between $3$ and $4$ .

Find 5 rational numbers between $\dfrac{2}{5}$ and $\dfrac{3}{5}$ .

Find 5 rational numbers between $\dfrac{1}{6}$ and $\dfrac{2}{5}$ .

If $\dfrac{x}{3} + \dfrac{x}{5} = \dfrac{16}{15}$ , find the rational number $x$ .

Let $a$ and $b$ be two non-zero rational numbers such that $a + \dfrac{1}{b} = 0$ . Without assigning any numerical values, determine whether $ab$ is positive or negative. Justify your answer.

Q10

A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4th decimal place. Show that such a number can be written in the form $\dfrac{p}{10^4}$ , where $p$ is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by $2^4$ or $5^4$ ? Give reasons.

Q11

Without performing division, determine whether the decimal expansion of $\dfrac{18}{125}$ is terminating or non-terminating. If it terminates, state the number of decimal places.

Q12

A rational number in its lowest form has denominator $2^3 \times 5$ . How many decimal places will its decimal expansion have? Explain your answer.

Q13

Let $a = \dfrac{7}{12}$ and $b = \dfrac{5}{6}$ . Express both $a$ and $b$ in the form $\dfrac{k_1}{m}$ and $\dfrac{k_2}{m}$ where $k_1$ , $k_2$ and $m$ are integers and $k_2 - k_1 > 6$ . Using the same denominator $m$ , write exactly five distinct rational numbers lying between $a$ and $b$ keeping an integer numerator. Explain why the condition $k_2 - k_1 > n + 1$ is necessary to find $n$ such rational numbers between the two rational numbers $a$ and $b$ using this method.

Q14

Three rational numbers $x, y, z$ satisfy $x + y + z = 0$ and $xy + yz + zx = 0$ . Show that all the rational numbers $x, y, z$ must be simultaneously zero.

Q15

Show that the rational number $\dfrac{(a+b)}{2}$ lies between the rational numbers $a$ and $b$ .

Q16

Find the lengths of the hypotenuses of all the right triangles in Fig. 3.14 which is referred to as the square root spiral.