NEW SYLLABUS TOPIC 2 - NUMBERS | BASIC MATHEMATICS FORM ONE (1)

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Concept of numbers

Numbers are classified into different categories. Some categories of numbers that you have already learned include whole numbers, natural numbers, fractions, integers, and decimals. Other major categories of numbers include rational, irrational, and real numbers. Activity 2.1 enables you to identify categories of numbers from daily life activities.

Activity 2.1: Categorising numbers

1. Use different measuring tools such as tape measure, ruler, weighing balance, and measuring cylinders to measure lengths, masses, and volumes of different objects, respectively. 2. Record the measurements in task 1 and categorise them based on your understanding of categories of numbers.

Quiz

Answer all questions.

1.

Which of the following is NOT a category of numbers mentioned in the passage?

1. A) Fractions
2. B) Natural numbers
3. C) Whole numbers
4. D) Imaginary numbers

Rational numbers

A rational number is any number that can be written in the form of \( \displaystyle \frac{a}{b} \) , where a and b are integers, and b ≠ 0. The condition b ≠ 0 is essential because division by zero is not defined. The set of rational numbers is denoted by the symbol ℚ.

Fractions, integers, whole numbers, terminating and non-terminating decimals form the set of rational numbers. For instance,

\( \displaystyle -\frac{34}{3} \), \( \displaystyle -\frac{20}{3} \), -3,  \( \displaystyle -\frac{4}{7} \),  \( \displaystyle -\frac{5}{6} \) , \( \displaystyle \frac{1}{2} \) , \( \displaystyle \frac{1}{3} \), \( \displaystyle \frac{4}{4} \), \( \displaystyle \frac{9}{5} \) , \( \displaystyle \frac{100}{5} \) , 0, \( \displaystyle \frac{30}{1} \) , 0.45, \( \displaystyle 0.\dot{3} \), \( \displaystyle 0.\dot{2} \dot{3} \), and \( \displaystyle 0.56\dot{7} \) are rational numbers.

Representation of rational numbers on a number line

Rational numbers can be represented on a number line. Positive rational numbers are represented on the right of zero (the origin) and the negative rational numbers on the left of the origin. The number line helps us to determine other rational numbers between any two rational numbers by increasing the number of divisions. Activity 2.2 enables you to locate numbers on a number line.

Activity 2.2: Locating numbers on a number line

1. Prepare a number line (2 to 3 metres long) using sticks and masking tapes or any other materials of your choice. 2. Cut manila cards in small sizes and write on them different types of numbers such as whole, integers, and fractions. 3. Give the cards to others to post each of the numbers on appropriate position on the number line. 4. While sticking the cards, explain to others why you think the positions are appropriate.

The representation of any positive rational number \( \displaystyle \frac{a}{b} \) is done by dividing the unit interval into ‘b’ equal parts. The ‘a’ of these parts are taken along the number line to reach the point corresponding to \( \displaystyle \frac{a}{b} \) on the right of zero if the number is positive and to the left of zero if the number is negative.

Example 2.1 Represent  \( \displaystyle \frac{1}{4} \) on a number line. Solution In order to represent \( \displaystyle \frac{1}{4} \) on a number line, take one unit from 0 towards the right side and then divide that unit into 4 equal parts. Take one part out of the 4 parts to complete a part representing \( \displaystyle \frac{1}{4} \). Therefore, \( \displaystyle \frac{1}{4} \) on a number line is represented as shown in the following number line.

Example 2.2 Represent - 2.4 on a number line. Solution - 2.4 = - \( \displaystyle 2\frac{4}{10} \)    = -\( \displaystyle 2\frac{2}{5} \) To represent −2.4 on a number line, take 2 units from 0 towards the left side and then divide the third unit into 5 equal parts.Take 2 parts out of the 5 parts to complete a part representing −2.4. Therefore,−2.4 on a number line is represented as shown in the following number line.

Example 2.3 Represent \( \displaystyle \frac{13}{5} \) and \( \displaystyle -\frac{13}{5} \) on the same number line. Solution Convert \( \displaystyle \frac{13}{5} \) into a mixed number. That is, \( \displaystyle \frac{13}{5}=2\frac{3}{5} \) \( \displaystyle =2+\frac{3}{5} \). To draw a number line, take 2 units from 0 towards the right side and then divide the third unit into 5 equal parts. Add the 2 units and the 3 parts out of the 5 parts to complete \( \displaystyle \frac{13}{5} \) . Also, convert \( \displaystyle -\frac{13}{5} \) into a mixed number, that is, \( \displaystyle -2\frac{3}{5} \). Repeat the same procedure from 0 to the negative side of the number line to represent \( \displaystyle -\frac{13}{5} \) . Therefore, \( \displaystyle \frac{13}{5} \) and \( \displaystyle -\frac{13}{5} \) on a number line are represented as shown in the following number line.

Exercise 2.1 1. Represent each of the following rational numbers on a number line. (a) 5 (b) \( \displaystyle \frac{7}{3} \)  (c)\( \displaystyle \frac{1}{5} \)  (d) 4.5    (e) \( \displaystyle \frac{3}{4} \) (f) \( \displaystyle \frac{3}{2} \)  (g) 0 (h)\( \displaystyle 3\frac{3}{5} \) 2. Show the position of each of the following rational numbers on a number line. (a) \( \displaystyle 2\frac{2}{3} \)  (b) \( \displaystyle \frac{6}{1} \) (c) \( \displaystyle \frac{16}{8} \) (d) \( \displaystyle \frac{0}{3} \)   (e) \( \displaystyle \frac{25}{6} \) (f)\( \displaystyle \frac{25}{50} \) (g) \( \displaystyle \frac{5}{5} \)   (h) \( \displaystyle \frac{50}{25} \) 3. Represent each of the following numbers on a number line.   (a) 3   (b) \( \displaystyle -\frac{5}{2} \)    (c) –3 4. (a) Write all negative and positive rational numbers from the following list of numbers. −5, 0, 12, −3.416520...,  2.85, 7.14, 12.64646464, \( \displaystyle \frac{11}{5} \), \( \displaystyle -\frac{3}{5} \), \( \displaystyle -2\frac{1}{3} \) . (b) Which numbers from (a) are not rational numbers? Justify your answer. 5. Write each of the following rational numbers in the form of in its simplest form. (a) 0.004  (b) 4  (c) −27 (d) 5.6 (e) −0.03 (f) \( \displaystyle 4\frac{1}{4} \)  (g) \( \displaystyle -6\frac{1}{8} \)   (h) \( \displaystyle -2\frac{2}{3} \) 6. Represent each of the following rational numbers on a number line. (a) −4    (e) \( \displaystyle -\frac{5}{3} \) (b) −2.5   (f) − 0.9, −0.8, 0.3, 0.2, 0.7 (c) 4     (g) \( \displaystyle -\frac{3}{10} \) (d) 1.8   (h) \( \displaystyle -\frac{3}{2} \), \( \displaystyle -\frac{5}{4} \), \( \displaystyle -\frac{1}{2} \), \( \displaystyle \frac{1}{4} \), \( \displaystyle \frac{1}{2} \), \( \displaystyle \frac{3}{2} \), \( \displaystyle \frac{5}{4} \), \( \displaystyle -\frac{3}{4} \) 7. Write 5 rational numbers which are equivalent to \( \displaystyle \frac{7}{10} \). 8. Write true or false in each of the following questions.   (a) All whole numbers are rational numbers.   (b) All integers are rational numbers.   (c) The only rational numbers which are equal to their reciprocals are 1 and −1. 9. Write down the rational number whose numerator is (−2 × 3) and whose denominator is (23−16) × (9−4). 10. A monthly salary of a certain worker is Tshs 250,000. One-fifth of his salary is spent in buying car fuels, half of remaining salary is spent in buying food and half of the rest is spent in miscellaneous expenses.   (a) How much is his monthly savings?   (b) State whether the answer in part (a) is a rational number or not.

Repeating decimals

Decimal numbers are part of rational numbers and are common in our daily life activities. The quantities such as length, height, age, volume and mass can be presented in decimals. Activity 2.3 guides you in expressing various quantities in fractions into decimals.

Activity 2.3: Expressing measurements of quantities in decimals

1. Take some fruits or similar objects and divide them into two, three, four, five, six, and seven equal parts. 2. Convert each fraction in task 1 into decimals in many decimal places as possible. You can work manually or use a calculator. 3. Study carefully the decimal part of the fractions and write down their unique characteristics.

A repeating decimal, also known as recurring decimal is a decimal number with at least one digit in the decimal part that repeats consecutively in a regular order without an end. For example, 1.6666666… and 0.639639639639… are repeating decimals because 6 and 639 digits from the two decimal numbers, respectively, in the decimal part repeat themselves without an end. The three dots indicate that the repeating digits continue infinitely.
Repeating decimals can also be represented by using a dot or a bar that is placed on top of a repeating digit.

Example 2.4 Write each of the following repeating decimals by using a dot and a bar. (a) 0.3333... (b) 0.639639639... (c) 0.474747474... Solution (a) 0.3333... is a repeating decimal which can be written as \( \displaystyle 0.\dot{3} \) or \( \displaystyle 0.\bar{3} \) (b) 0.639639639... can be written as \( \displaystyle 0.\dot{6}3\dot{9} \) or \( \displaystyle 0.\overline{639} \) (c) 0.474747474... can be written as \( \displaystyle 0.\dot{4}\dot{7} \) or \( \displaystyle 0.\overline{47}\)

In Example 2.4, the digits with a dot or a bar are repeating infinitely. In (b) and (c), it can be observed that if a group of digits is repeating, a dot should be put over the first and the last repeating digits.
Decimals are either terminating or non-terminating. Terminating decimals have a definite number of digits after the decimal point while non-terminating decimals have an endless number of digits after the decimal point. Thus, repeating decimals are non-terminating with one or more repeating digits in the decimal part.
Examples of terminating decimals are 0.5, 1.4, and 7.9 while non-repeating decimals are 3.1415926..., 1.4142135…., and 2.2360679…

Converting repeating decimals into fractions

When working with problems involving repeating decimals, it is important to convert them into simple fractions to maintain accuracy and avoid errors. A repeating decimal can be converted into fraction using the following steps:

1. Choose any variable to represent the required fraction.

2. Multiply both sides of the equation by a multiple of 10 depending on the number of repeating decimals. For example, \( \displaystyle 0.\dot{8}\) and \( \displaystyle 0.21\dot{3}\) have only one repeating digit which means they are multiplied by 10. Decimal numbers \( \displaystyle 0.\dot{1}\dot{1}\) and \( \displaystyle 1.2\dot{1}\dot{4}\) have two repeating digits, thus they are multiplied by 100, and \( \displaystyle 0.\dot{8}3\dot{5}\) will be multiplied by 1,000 since it has 3 repeating digits.

3. Subtract the equation in step 1 from the equation in step 2.

4. From the equation obtained in step 3, solve for the chosen variable and simplify where necessary.

Example 2.5 Convert each of the following decimals into fractions. (a)\( \displaystyle 0.\dot{3}\)   (b) \( \displaystyle 0.\dot{8}\dot{3}\)   (c) \( \displaystyle 0.\dot{8}3\dot{5}\)    (d) \( \displaystyle 0.8\dot{3}\) Solution (a) Let x =\( \displaystyle 0.\dot{3}\)..........(i) Multiply by 10 both sides of equation (i) to obtain, 10x = \( \displaystyle 3.\dot{3}\)...............(ii) Subtract equation (i) from equation (ii) as follows: 10x − x = \( \displaystyle 3.\dot{3}\) − \( \displaystyle 0.\dot{3}\) Thus, 9x = 3     x = \( \displaystyle \frac{1}{3} \) Therefore, \( \displaystyle 0.\dot{3}\) into fraction is \(\displaystyle \frac{1}{3}\) . (b) \( \displaystyle 0.\dot{8}\dot{3}\) Let x = \( \displaystyle 0.\dot{8}\dot{3}\)...............(i) Multiply by 100 both sides of equation (i) to obtain 100x =  \( \displaystyle 83.\dot{8}\dot{3}\).........(ii) Subtract equation (i) from equation (ii) as follows: 100x − x = \( \displaystyle 83.\dot{8}\dot{3}\) − \( \displaystyle 0.\dot{8}\dot{3}\)              99x =  83                x = \( \displaystyle \frac{83}{99}\) Therefore, \( \displaystyle 0.\dot{8}\dot{3}\) into fraction is \(\displaystyle \frac{83}{99}\) . (c) \( \displaystyle 0.\dot{8}3\dot{5}\) Let x = \( \displaystyle 0.\dot{8}3\dot{5}\) ..............(i) Multiply by 1 000 both sides of equation (i) to get, 1000x = \( \displaystyle 835.\dot{8}3\dot{5}\) ........(ii) Subtract equation (i) from equation (ii) and solve for x as follows: 1000x − x = \( \displaystyle 835.\dot{8}3\dot{5}\) − \( \displaystyle 0.\dot{8}3\dot{5}\)                999x = 835            x = \( \displaystyle \frac{835}{999} \) Therefore, \( \displaystyle 0.\dot{8}3\dot{5}\) into a fraction is \(\displaystyle \frac{835}{999}\) . (d) \( \displaystyle 0.8\dot{3}\) Let x = \( \displaystyle 0.8\dot{3}\) ........(i) Multiply by 10 both sides of equation (i) to get, 10x = \( \displaystyle 8.3\dot{3}\) ..........(ii) Subtract equation (i) from equation (ii) and solve for x as follows: 10x − x =\( \displaystyle 8.3\dot{3}\) − \( \displaystyle 0.8\dot{3}\)         9x = 7.5         x = \(\displaystyle \frac{7.5}{9}\)         = \(\displaystyle \frac{5}{6}\) Therefore, \( \displaystyle 0.8\dot{3}\) into fraction is \(\displaystyle \frac{5}{6}\).

Example 2.6 Convert each of the following decimals into fractions. (a) \( \displaystyle 9.\dot{2}\) (b) \( \displaystyle 5.\dot{2}\dot{3}\) (c) \( \displaystyle 2.\dot{1}0\dot{5}\) Solution (a) \( \displaystyle 9.\dot{2}\) Let x = \( \displaystyle 9.\dot{2}\)...............(i) Multiply by 10 both sides of equation (i) to get, 10x = \( \displaystyle 92.\dot{2}\)..............(ii) Subtract equation (i) from equation (ii) and solve for x as follows: 10x − x = \( \displaystyle 92.\dot{2}\) − \( \displaystyle 9.\dot{2}\) Thus, 9x = 83   x = \(\displaystyle \frac{83}{9}\) Therefore, \( \displaystyle 9.\dot{2}\) into fraction is \(\displaystyle \frac{83}{9}\). (b) \( \displaystyle 5.\dot{2}\dot{3}\) Let x = \( \displaystyle 5.\dot{2}\dot{3}\) .............(i) Multiply by 100 both sides of equation (i) to get 100x =  \( \displaystyle 523.\dot{2}\dot{3}\) .....(ii) Subtract equation (i) from equation (ii) and solve for x as follows: 100x − x = \( \displaystyle 523.\dot{2}\dot{3}\) − \( \displaystyle 5.\dot{2}\dot{3}\)           99x = 518           x = \(\displaystyle \frac{518}{99}\) Therefore, \( \displaystyle 5.\dot{2}\dot{3}\) into fraction is \(\displaystyle \frac{518}{99}\) . (c) \( \displaystyle 2.\dot{1}0\dot{5}\) Let x = \( \displaystyle 2.\dot{1}0\dot{5}\).....................(i) Multiply by 1 000 both sides of equation (i) to get, 1000x = \( \displaystyle 2105.\dot{1}0\dot{5}\)........(ii) Subtract equation (i) from equation (ii) and solve for x as follows: 1000x − x = \( \displaystyle 2105.\dot{1}0\dot{5}\) − \( \displaystyle 2.\dot{1}0\dot{5}\)              999x = 2103              x = \(\displaystyle \frac{2103}{999}\)               = \(\displaystyle \frac{701}{333}\) Therefore, \( \displaystyle 2.\dot{1}0\dot{5}\) into a fraction is \(\displaystyle \frac{701}{333}\) .

Exercise 2.2 1. Convert each of the following decimals into fractions. (a) \( \displaystyle 0.\dot{1}\) (b) \( \displaystyle 0.\dot{8}\) (c) \( \displaystyle 80.\dot{2}1\dot{7}\) (d) \( \displaystyle 3.1\dot{1}\dot{2}\) (e) \( \displaystyle 13.0\dot{1}\dot{5}\) (f) \( \displaystyle 0.3\dot{4}\) (g) \( \displaystyle 0.\dot{7}2\dot{3}\)  (h) \( \displaystyle 2.\dot{4}\)      2. Which of the following rational numbers are repeating decimals?  (a) \(\displaystyle \frac{1}{9}\) (b) \(\displaystyle \frac{22}{7}\) (c) \(\displaystyle 3\frac{2}{3}\) (d) \(\displaystyle \frac{20}{11}\) 3. If x = \( \displaystyle 2.\dot{6}\) and y = \( \displaystyle 2.8\dot{3}\), find the value of \(\displaystyle \frac{x}{y}\). 4. If x =\( \displaystyle 0.\dot{3}\) and y = \( \displaystyle 0.\dot{6}\), verify that y2 = x2 + x. 5. Evaluate the following giving your answer as a fraction in its simplest form.  (a) \( \displaystyle 0.\dot{2}\) + \( \displaystyle 0.\dot{5}\dot{1}\) (b) \( \displaystyle 1.6\dot{7}\) + \( \displaystyle 2.\dot{1}\) (c) \( \displaystyle 0.6\dot{6}\) − \( \displaystyle 0.4\dot{1}\) 6. Show that \( \displaystyle 0.\dot{4}\dot{6}\) = \(\displaystyle \frac{46}{99}\) . 7. Evaluate the following and give your answer in fractions.     \( \displaystyle 1.2\dot{5}\dot{6}\) + \( \displaystyle 0.\dot{7}\) – 0.15 8. The fraction, \(\displaystyle a\frac{b}{c}\) is equivalent to \( \displaystyle 5.\dot{1}6\dot{5}\).  Find the values of a, b, and c. 9. What is the sum of the numerator and denominator of \( \displaystyle 0.\dot{2}\dot{7}\) when converted into fraction? 10. Why learning the concept of recurring decimals is important in your daily life?

Converting fractions into repeating decimals

A fraction can be converted into a decimal by performing a long division. In this process, some fractions will be equivalent to either terminating or non-terminating decimals. If the resulting decimal is a non-terminating with recurring decimals, the repeating digits are indicated by the repeating decimal’s notation.

Example 2.7 Convert the following fractions into decimals. (a) \(\displaystyle \frac{1}{3}\) (b) \(\displaystyle \frac{2}{3}\) (c) \(\displaystyle \frac{5}{33}\) (d) \(\displaystyle \frac{17387}{909}\) Solution (a) Divide 1 by 3 using long division as follows and check your answer by a calculator.   Therefore,\(\displaystyle \frac{1}{3}\) = \(\displaystyle 0.\dot{3}\) (b) Divide 2 by 3 using long division as follows and check your answer by a calculator. Therefore, \(\displaystyle \frac{2}{3}\) = \(\displaystyle 0.\dot{6}\) (c) Divide 5 by 33 by long division or by a calculator to get 0.151515151515… . Note that in the answer, 1 and 5 repeats as you proceed to divide. Therefore, \(\displaystyle \frac{5}{33}\) = \(\displaystyle 0.\dot{1}\dot{5}\) or \(\displaystyle 0.\overline{15}\). (d) Divide \(\displaystyle \frac{17387}{909}\) by long division or by a calculator to obtain 19.127612761276… Note that in the answer, the digits 1, 2, 7, and 6 repeats as you proceed to divide. Therefore, \(\displaystyle \frac{17387}{909}\) = \(\displaystyle 19.\dot{1}27\dot{6}\) or \(\displaystyle 19.\overline{1276}\).

Example 2.8 1. Convert each of the following mixed fractions into decimals. (a) \(\displaystyle 1\frac{5}{9}\)     (b) \(\displaystyle1\frac{37}{45}\)    (c) \(\displaystyle 9\frac{1}{6}\) Solution (a) Given \(\displaystyle 1\frac{5}{9}\). Converting the mixed fraction into improper fraction gives, \(\displaystyle 1\frac{5}{9}\) = \(\displaystyle \frac{14}{9}\) Dividing 14 by 9 using the long division method gives \(\displaystyle 1.\dot{5}\). Therefore, \(\displaystyle 1\frac{5}{9}\) = \(\displaystyle 1.\dot{5}\). (b) Given \(\displaystyle 1\frac{37}{45}\). Converting the mixed fraction into improper fraction gives, \(\displaystyle 1\frac{37}{45}\) = \(\displaystyle \frac{82}{45}\) Dividing 82 by 45 using the long division method gives \(\displaystyle 1.8\dot{2}\). Therefore, \(\displaystyle 1\frac{37}{45}\) = \(\displaystyle 1.8\dot{2}\). (c) Given \(\displaystyle 9\frac{1}{6}\). Converting the mixed fraction into improper fraction gives, \(\displaystyle 9\frac{1}{6}\) = \(\displaystyle \frac{55}{6}\) . Divide 55 by 6 using the long division method to get \(\displaystyle 9.1\dot{6}\). Therefore, \(\displaystyle 9\frac{1}{6}\) = \(\displaystyle 9.1\dot{6}\).

Example 2.9 A wire is divided into two pieces of lengths \(\displaystyle 2\frac{1}{3}\) cm and \(\displaystyle 2\frac{1}{2}\) cm. Find the total length of the wire in decimals. Solution Given the lengths of the two pieces of wire as \(\displaystyle 2\frac{1}{3}\) cm and \(\displaystyle 2\frac{1}{2}\) cm. Add the mixed fractions as follows: \(\displaystyle 2\frac{1}{3}\) + \(\displaystyle 2\frac{1}{2}\) = \(\displaystyle \frac{7}{3}\) + \(\displaystyle \frac{5}{2}\)       = \(\displaystyle \frac{29}{6}\) Divide 29 by 6 using the long division method to obtain \(\displaystyle 4.8\dot{3}\). Therefore, the total length of the wire is \(\displaystyle 4.8\dot{3}\) cm.

Exercise 2.3 1. Convert each of the following fractions into decimals. (a) \(\displaystyle \frac{6}{7}\)   (b) \(\displaystyle \frac{11}{6}\)  (c) \(\displaystyle \frac{8}{11}\)   (d) \(\displaystyle \frac{102}{90}\)  (e) \(\displaystyle \frac{25}{13}\) (f) \(\displaystyle \frac{21}{13}\) 2. Which of the following rational numbers are repeating decimals? (a) \(\displaystyle \frac{1}{9}\)   (b) \(\displaystyle \frac{22}{7}\)  (c) \(\displaystyle 3\frac{2}{3}\) (d) \(\displaystyle \frac{20}{11}\) 3. Which is greater between \(\displaystyle 0.\dot{1}\dot{2}\) and \(\displaystyle \frac{1}{3}\) ? 4. Write the following numbers in ascending order: \(\displaystyle \frac{11}{23}\) , \(\displaystyle 0.4\dot{7}\dot{2}\), and \(\displaystyle \frac{5}{11}\). 5. Show that \(\displaystyle \frac{23}{9}\) = 2.55555… 6. Mwajuma has \(\displaystyle \frac{1}{3}\) of a chocolate bar. She decides to share it equally among her 5 friends. How much chocolate will each friend get? Write your answer in fraction and decimal. 7. John wants to divide a 1 metre-long ribbon so that he can share it equally with his two friends. How much in metres will each get? Express your answer in decimals. 8. If m = \(\displaystyle 0.\dot{2}\) and n = \(\displaystyle 0.\dot{0}\dot{4}\),  show that m2 = n(m + 1). 9. A car travels 2/3 of the total distance to its destination at a speed of 30 km/h, and then travels the remaining distance at a speed of 40 km/h. If the car took 4 hours for the whole journey, what is the total distance of the journey? Express your answer as a repeating decimal. 10. The base of a lid of bucket has an area of \(\displaystyle \frac{25}{484}\)π m2. Find its diameter in decimal.

Quiz

Answer all questions.

1.

A rational number can always be written in the form of:

1. A) ${\displaystyle \frac{a}{b}}$ where is a an integer and b is a decimal
2. B) ${\displaystyle \frac{a}{b}}$ where a and b are integers and b ${\displaystyle \ne }$0
3. C) ${\displaystyle \frac{a}{b}}$ where both a and b are negative numbers
4. D) ${\displaystyle \frac{a}{b}}$ where and are integers and b=0

Irrational numbers

Irrational numbers are special numbers in our life. A widely known and used irrational number is Pi (π) which appears in formulas for determining circumferences, areas, and volumes of circular shapes. Activity 2.4 allows you to use your experience in decimals to learn about irrational numbers.

Activity 2.4: Differentiating types of decimal numbers

1. Identify 10 different fractions which can be converted into terminating or repeating decimals. 2. Use a calculator to find answers to the square roots of at most 10 numbers that have no perfect squares and write your answers with at least 10 decimal places. 3. Compare the answers in tasks 1 and 2 and write down the differences observed.

An irrational number is a number which can be written as a non-terminating and non-repeating decimal. Also, these numbers cannot be expressed in the form of  \(\displaystyle \frac{a}{b}\), where a and b are integers and b ≠ 0. The set of irrational numbers is denoted by ℚ′. Irrational numbers cannot be represented exactly on a number line. However, they can always be approximated to rational numbers.

Example 2.10 Write any 6 irrational numbers. Solution The following is a list of some irrational numbers. (i) \(\displaystyle \sqrt{2}\) = 1.414213562373…       (iv) \(\displaystyle -\sqrt{5}\) = − 2.236067977499… (ii) e = 2.718281828459…      (v) \(\displaystyle \sqrt{11}\) = 3.316624790355… (iii) π = 3.141592653589…     (vi) \(\displaystyle - \sqrt{21}\) = − 4.582575694956... From these examples, it can be observed that the digits after the decimal point continue infinitely without repeating.

Note: The numbers such as π, e, eπ, 2π, e + π, and other similar numbers are also called transcendentals.

Example 2.11 Given the list of numbers -6,  \(\displaystyle -5\frac{3}{5}\), \(\displaystyle -\sqrt{4}\),  \(\displaystyle -\frac{3}{5}\), \(\displaystyle -\frac{3}{8}\), 0, \(\displaystyle \frac{4}{5}\), 1, \(\displaystyle 1\frac{2}{3}\) , \(\displaystyle \sqrt{8}\), 3.01,  π,  8.47. Identify all irrational numbers in the list. Solution The irrational numbers are \(\displaystyle \sqrt{8}\) and π.

Exercise 2.4 1. Determine whether the following are rational or irrational numbers. Justify your answers. \(\displaystyle \sqrt{11}\), 0.65, \(\displaystyle \sqrt{12}\), \(\displaystyle \frac{\pi}{2}\), 1.101001000100001…, \(\displaystyle \sqrt{9}\), \(\displaystyle 3+\sqrt{7}\), \(\displaystyle 3\sqrt{3}-\sqrt{5}\),   \(\displaystyle -\sqrt{7}\) 2. Find an irrational number such that x2 = 7. 3. Given the numbers -5, -3.5,  0, \(\displaystyle \frac{3}{4}\),  \(\displaystyle \sqrt{3}\), \(\displaystyle \sqrt{5}\), 9, \(\displaystyle -4.3\dot{2}\dot{5}\), \(\displaystyle \pi\),  \(\displaystyle \frac{\sqrt{2}}{2}\). List the numbers which are: (a) Whole numbers                  (c) Rational numbers (b) Integers               (d) Irrational numbers 4. State whether the following statements are true or false. Justify your answers.    (a) Every irrational number is a real number.    (b) All non-terminating decimals are irrational numbers.    (c) The square roots of all positive numbers are irrational numbers.    (d) The sum of two irrational numbers gives an irrational number. 5. Ramadhani is planning to build a circular pond whose base area is 100 square metres. Express the radius of the base of the pond as an irrational number. 6. Identify irrational numbers from the following list of numbers: \(\displaystyle \sqrt{9}\), 0.54, \(\displaystyle \frac{91}{13}\), \(\displaystyle 1.2\dot{5}\dot{6}\), \(\displaystyle \frac{\sqrt{2}}{2}\) , \(\displaystyle \frac{1}{11}\). 7. (a) What is an irrational number?  (b) Explain the fact that the sum of two or more irrational numbers gives an irrational number. 8. Provide an example for each of the following statements.     (a) The product of rational and irrational number is an irrational number.     (b) The product of two irrational numbers is an irrational number. 9. Write five irrational numbers with denominators as irrational numbers. 10. In the following list of numbers, identify a list which does not contain irrational numbers.      (a) 0.1, \(\displaystyle \frac{2}{7}\), 5.33333…      (b) \(\displaystyle \sqrt{16}\), \(\displaystyle \sqrt{1}\), \(\displaystyle \sqrt{9}\), 3.844231028...      (c) \(\displaystyle \sqrt{4}\),  \(\displaystyle 4\frac{2}{9}\),   0.6,   7      (d) \(\displaystyle -\frac{28}{9}\), 0, \(\displaystyle \sqrt{25}\),  3.45

Quiz

Answer all questions.

1.

Your friend is struggling to understand irrational numbers. Which explanation is correct?

1. A) They are numbers that repeat and terminate
2. B) These are numbers that cannot be written as fractions
3. C) They include only whole numbers
4. D) They have exact decimal values

Real numbers

Real numbers are the numbers which include both rational and irrational numbers. A set of real numbers is denoted by ℝ. Thus, all sets of numbers such as natural numbers, whole numbers, integers, rational numbers and irrational numbers are all real numbers. Natural numbers are the smallest set of numbers followed by whole numbers and integers which are all rational numbers. Rational and irrational numbers are opposite sets of numbers and are what makes the largest set of real numbers as shown in Figure 2.1.

 

Figure 2.1: Categories of real numbers

If a and b are two real numbers, then either a = b, a is less than b or a is greater than b. For example, if:

(i) a = 3 and b = 7, then a is less than b or b is greater than a, that is, 3 is less than 7 or 7 is greater than 3.

(ii) a =  \(\displaystyle \frac{12}{3}\) and b = 4, then, a = b that is,  \(\displaystyle \frac{12}{3}\) = 4.

(iii) a = −3 and b = −5, then a is greater than b, that is −3 is greater than −5 or b is less than a, that is, −5 is less than −3.

(iv) a = –4 and b = 2 then a is less than b, that is – 4 is less than 2 or b is greater than a, that is, 2 is greater than −4.

Examples of positive real numbers are 1, 2, 3,  \(\displaystyle \frac{2}{3}\), \(\displaystyle \sqrt{7}\), 1.67, and 1.020020002…  and examples of negative real numbers are: -1, -2, -3, \(\displaystyle -\frac{3}{2}\), \(\displaystyle -\sqrt{7}\), -1.067, -1.020020002…

Note: 1. 0 is also a real number which is neither positive nor negative.       2. Every real number corresponds to a single point on the number line.       3. Every point on the number line corresponds to a certain real number.

Quiz

Answer all questions.

1.

Which of the following statements about real numbers is correct?

1. A) Real numbers do not include zero
2. B) Real numbers include only positive numbers
3. C) Real numbers are only whole numbers
4. D) Every real number is either a rational or an irrational number

Inequalities in real numbers

In daily life, people are faced with problems related to comparing quantities of the same item for the purpose of making decisions. Consider the following examples:

(i) In some places, the speed of the car is limited to a certain maximum value due to the large number of pedestrians.

(ii) Event organisers can set a maximum number of attendees for the event.

(iii) Most banks limit the withdrawal amount to a certain minimum and maximum amount per day in automated teller machines.

(iv) In schools, a minimum pass mark is set for students to be awarded certificates of completion of education.

Activity 2.5 guides you in comparing quantities in real life.

Activity 2.5: Comparing quantities in real life

1. Measure the lengths, widths, and heights of different objects of your choice. 2. Compare the measured values of a pair of objects in task 1.

In Mathematics, an inequality is a statement that compares or relates two values or expressions. The common terms involved in comparing the values or expressions are “less than”, “greater than”, “greater than or equal to”, “less than or equal to”, or “not equal to”. For instance, the following statements are examples of inequalities:

(i) −5 is less than 0.

(ii) John’s age is greater than Jane’s age.

(iii) x is less than or equal to 9.

(iv) y is greater than or equal to 0.

(v) 2.5 is not equal to 1.2.

The mathematical statements can be written using symbols which represent inequalities as shown in the following table.

Word	Inequality symbols	Example
Less than	<	- 5 < 0
Greater than	>	John’s age > Jane’s age
Less than or equal	≤	x ≤ 9
Greater than or equal	≥	y ≥ 0
Not equal to	≠	2.5 ≠ 1.2

In general, mathematical statements which use ≠, >, < ,≥, or ≤ are called inequalities.

Note: 1. The sharp end of > and < always points to a smaller number. 2. The symbols ≤ and ≥ are sometimes referred to as at most and at least,respectively. 3. If the inequality is divided or multiplied both sides by a negative number, the direction of the inequality sign changes.

Suppose a, b, and c are real numbers which are represented on the number line shown in the following number line:

Generally, on a number line, a number to the right side of another number is always greater than a number to its left side. Thus, from the number line, 

b is greater than a, which can also be written as b > a.

c is greater than b, which can also be written as c > b.

The vice versa is also true, that is, a number which is on the left side of another number on a number line is always less than the number on its right side. That is, a < b and b < c.

Example 2.12 Study the following number line and use inequality symbols to answer the questions that follow. (a) Write all real numbers which are greater than zero. (b) Write all real numbers which do not exceed 2. (c) Write all real numbers between –3 and 4. Solution (a) Let x be any real number that is greater than 0. The symbol > is used to represent the solution. There are infinity real numbers which are greater than zero. Therefore, x > 0. (b) Let x be any real number which does not exceed 2. Numbers on the left side of 2 on a number line are always less than 2. Also, 2 is included as it satisfies the given condition. Thus, the less than or equal inequality symbol should be used to present the solution. Therefore, x ≤ 2. (c) Let x be a real number between −3 and 4. This means that −3 < x and x < 4. Combining the two inequalities gives, −3 < x < 4.

Example 2.13 If x is an integer, locate each of the following on a number line. (a) −1 is less than x and x is less than 4. (b) −2 is less than or equal to x and x is less than or equal to 6. Solution (a) The statement is equivalent to −1 < x < 4, where x is an integer. On a number line, it is represented as follows: (b) The statement is equivalent to −2 ≤ x ≤ 6, where x is an integer. On a number line, it is represented as follows:

Example 2.14 Use a number line to compare each of the following pair of numbers. (a) 0.54 and 0.33 (b) −0.54 and −0.33 Solution (a) 0.54 and 0.33 are represented in the following number line. From the number line, 0.33 is on the left side of 0.54, which implies that it is less than 0.54. Therefore, 0.33 < 0.54 or 0.54 > 0.33. (b) –0.54 and –0.33 are located on a number line as follows: From the number line, −0.33 is on the right side of −0.54, hence −0.33 is greater than −0.54. Therefore, −0.33 > −0.54 or −0.54 < −0.33.

Example 2.15 Compare the following numbers by using the inequality symbols < and >. (a) 0.432 and 0.437        (c) 3.7244 and 3.724 (b) −0.127 and 0.001       (d) \(\displaystyle -\sqrt{7}\) and \(\displaystyle \sqrt{5}\) Solution (a) 0.432 < 0.437 or 0.437 > 0.432 (b) −0.127 < 0.001 or 0.001 > −0.127  (c) 3.7244 > 3.724 or 3.724 < 3.7244 (d) \(\displaystyle \sqrt{5}\) > \(\displaystyle -\sqrt{7}\) or \(\displaystyle -\sqrt{7}\) < \(\displaystyle \sqrt{5}\)

Exercise 2.5 1. Write true or false in each of the following inequalities.    (a) 3 < 30        (d) 6 ≥ 2                          (b) 5 > −2        (e) −3 < −2           (c) 3 < −30       (f) 2 > − 4        (g) −8 ≥ −4       (h) −4 < −5    (i) 5 < 10        (j) 8 = −8 2. Which of the following inequalities are true?    (a) −250 < −150        (d) 6 ≠ 6    (b) 2 ≤ −2             (e) -3 = 3     (c) 5 > −2             (f) 5 ≥ −2 3. List the numbers which satisfy each of the following conditions.    (a) x < 6 if x is a counting number.    (b) x ≤ 4 if x is a whole number.    (c) x > 3 if x is an odd number.    (d) x > −3 and x ≤ 4 if x is an integer. 4. In each of the following, insert a correct inequality symbols in the box to make a true statement.   (a) -5×(-3)\(\displaystyle \large \square \)8×(-3)        (b) 13 × (−2)\(\displaystyle \large \square\)7 × (−2)     (c) 8 × (5)\(\displaystyle \large \square\)2 × (−5)    (d) −6 × (−4)\(\displaystyle \large \square\)−3 × (−4)  (e) If a > b, then a × (−9)\(\displaystyle \large \square\)b × (−9) 5. Write all whole numbers in the interval −4≤ x < 6. 6. Give a verbal statement equivalent to each of the following inequalities.   (a) 1 < x ≤ 10    (b) x < −3 or x > 4. 7. Ester is shorter than Nancy and Nancy is taller than Hyasinta. Express mathematically the relationship between their heights. 8. Anna has Tshs 500,000 in her bank account. Every week she withdraws Tshs 40,000 for various expenses. Without making any deposit, how many weeks can she withdraw this money if she wants to maintain a balance of at least Tshs 200,000? 9. The sum of two consecutive integers is at least negative fifteen. What are the smallest values of consecutive integers that will make this true? 10. Represent mathematically three real-life inequality scenarios of your choice.

Representing real numbers on a number line

Sometimes it is not easy to list all elements of a set of real numbers. For example, it is not possible to list all real numbers which are greater than \(\displaystyle \frac{1}{2}\). However, the use of inequality symbols helps to define the elements in the given set. In this case, if x is a real number which is greater than \(\displaystyle \frac{1}{2}\), then x > \(\displaystyle \frac{1}{2}\) defines all the required elements in the set.

The inequality x > \(\displaystyle \frac{1}{2}\) means that there are infinitely many real numbers satisfying this condition. Thus, they can be represented on a number line by using an arrow with a small circle at  \(\displaystyle \frac{1}{2}\) (initial point) since it is not included in the set.

The following number line shows all the real numbers which are greater than \(\displaystyle \frac{1}{2}\).

Note: 1. A small circle ( \(\displaystyle \huge \circ \)) indicates that the number in a given set of real numbers is excluded in the given set. This is always true for a set of real numbers which involve < or > inequality symbols. 2. A thick dot (\(\displaystyle \huge \bullet \)) indicates that the number in a given set of numbers is included in the given set. This is always true for set of real numbers which involve ≤ or ≥ inequality symbols.

Example 2.16 Use a number line to represent all real numbers x in each of the following: (a) \(\displaystyle -\frac{1}{2}\)≤ x < 2     (c) \(\displaystyle -\frac{5}{3}\) < x ≤ 2 (b) x ≤ \(\displaystyle \frac{3}{2}\)        (d) x ≤ 3 or x ≥ 3 Solution

Example 2.17 Write the inequalities represented by each of the following number lines. Solution (a) −2 ≤ x ≤ 2  (b) \(\displaystyle -\frac{11}{4}\) < x ≤ \(\displaystyle \frac{3}{2}\) (c) x < − 1 or x ≥ 2  (d) −1 < x < 2

Example 2.18 Represent on a number line all the real numbers between \(\displaystyle -0.\dot{3}\)  and 2. Solution Let x be a set of real numbers between \(\displaystyle -0.\dot{3}\)  and 2. This set of real numbers can be represented as \(\displaystyle -0.\dot{3}\) < x < 2 since, both \(\displaystyle -0.\dot{3}\) and 2 are not included in the set. The number line representing this set of real numbers is shown in the following number line.

Exercise 2.6 1. Represent each of the following numbers on a number line.    (a) −4, −2, 0, 2, 4    (b) −2 ≤ k ≤ 1 where k is a real number    (c) −5 < y ≤ −2, where y is a real number    (d) −2 < x < 4, where x is a real number 2. Represent each of the following integers on a number line under the given conditions.  (a) −6 < a < −2    (c) c < 5 and c > −3    (b) −4  < b < −1  (d) −3  ≤ e ≤ 5  (e) f > −9 and f  ≤ −4. 3. Draw a number line to represent each of the following real numbers.   (a) x < −1 or x >  \(\displaystyle \frac{1}{2}\)   (b) −6 ≤ x and x ≤ −2 4. Use a number line to determine the distance between −3 and 2. 5. Represent the given set of real numbers on a number line: 0.2, 0.4, 0.6, and 0.8. 6. Locate three consecutive integers on a number line which are immediate greater than −5. 7. Represent the following numbers on a number line.    0.2, 0.4, −0.6, \(\displaystyle \frac{23}{5}\) , 6, \(\displaystyle \frac{32}{7}\) 8. Represent each of the following inequalities on a number line.  (a) \(\displaystyle -\frac{5}{2}\) < x < \(\displaystyle -\frac{1}{2}\)                    (b) \(\displaystyle\frac{1}{2}\) ≤ x < \(\displaystyle\frac{7}{2}\)              (c) \(\displaystyle -\frac{2}{3}\) < x ≤ 4    (d) x < \(\displaystyle -\frac{5}{2}\) or x > \(\displaystyle -2\frac{1}{3}\)   (e) 2 < x < \(\displaystyle 5\frac{1}{4}\)   (f) −3 ≤ x ≤ 4 9. Represent each of the following information on a number line.   (a) The expected mass of a human at birth is at least 3 kg and less than 5 kg.   (b) A manufacturing company produces buckets of at least 20 cm3.   (c) The mass of students in a class is at most 50 kg and not less than 35 kg. 10. On a certain number line, numbers are placed at an interval of 1 unit.    (a) What number is 6 units to the right of −4?    (b) What number is 3 units to the left of 2?

Activity 2.6 guides you in representing real numbers on a number line by using graphing tools.

 

Activity 2.6: Representing real numbers on a number line by using graphing tools

1. Identify different mathematical software which can be used in drawing and representing real numbers on a number line. 2. Study the basic procedures for drawing and representing numbers by using the software. 3. Use the software to draw and locate real numbers on a number line.

Quiz

Answer all questions.

1.

Which of the following is true about the inequality −3<x<4?

1. A) x must always be between −3 and 4, but not equal to them.
2. B) x can be less than −3 or greater than 4.
3. C) x can be any value greater than −3 or less than 4.
4. D) x can only be equal to −3 or 4.

Absolute value of a real number

The absolute value of a real number x, denoted by \(\displaystyle \left| x \right|\) is the non-negative value of x regardless of its sign. For example \(\displaystyle \left| 2 \right|\) = 2 and \(\displaystyle \left| -2 \right|\) = 2. Consider the following number line.

 

The distance from 0 to −2 and that from 0 to 2 is the same, that is 2 units. Thus, 2 is an absolute value of both 2 and −2. The absolute value of a number is the magnitude of that number regardless of its sign.

The absolute value of a number x is written as \(\displaystyle \left| x \right|\) such that \(\displaystyle \left| x \right|\) =x if x ≥ 0 and \(\displaystyle \left| x \right|\) = -x if x <0, and its distance is represented on a number line from 0. This distance is always positive or zero.

For example; \(\displaystyle \left| 0.75 \right|\) = 0.75,  \(\displaystyle \left| -0.143 \right|\) = 0.143, \(\displaystyle \left| -\frac{1}{2} \right|\) =  \(\displaystyle \frac{1}{2}\), \(\displaystyle \left| 0 \right|\) = 0, and \(\displaystyle \left| -2 \right|\) = 2.

In general, for any two positive and negative numbers x and -x, their absolute values are equal, that is, \(\displaystyle \left| x \right|\) = \(\displaystyle \left| -x \right|\).

Example 2.19 Evaluate each of the following: (a) \(\displaystyle  \left| -2 \right|\)     (b) \(\displaystyle \left| 10  \right|\)     (c) \(\displaystyle \left| 0 \right|\)     (d) \(\displaystyle \left| 8-2 \right|\) (e) \(\displaystyle \left| -3\times6 \right|\)  (f) \(\displaystyle \left| 2-6 \right|\)     (g) \(\displaystyle \left| 5+2 \right|\)             Solution (a)\(\displaystyle \left| -2 \right|\)=2   (b) \(\displaystyle \left| 10 \right|\)=10 (c) \(\displaystyle \left| 0 \right|\)=0   (d) \(\displaystyle \left| 8-2 \right|\) = \(\displaystyle \left| 6 \right|\) =6   (e) \(\displaystyle \left| -3\times6  \right|\)=\(\displaystyle \left| -18\right|\)=18 (f) \(\displaystyle \left| 2-6 \right|\)=\(\displaystyle \left| -4 \right|\)=4 (g) \(\displaystyle \left| 5+2 \right|\)=\(\displaystyle \left| 7 \right|\)=7

Example 2.20 Find the absolute value of  \(\displaystyle -\frac{13}{10}+\frac{4}{5}\)   Solution \(\displaystyle \left| -\frac{13}{10}+\frac{4}{5} \right|\)=\(\displaystyle \left| -\frac{13+8}{10} \right|\)          =\(\displaystyle \left| -\frac{5}{10} \right|\)          =\(\displaystyle \left| -\frac{1}{2} \right|\)          = \(\displaystyle \frac{1}{2}\) Therefore, the absolute value of \(\displaystyle -\frac{13}{10}+\frac{4}{5}\) is \(\displaystyle \frac{1}{2}\).

Example 2.21 Represent the absolute value of −2 on a number line. Solution \(\displaystyle \left| -2 \right|\)=2 . Thus, the representation of 2 on a number line is shown in the following number line.

Exercise 2.7 1. Find the absolute value in each of the following: (a) 62  (b) \(\displaystyle \sqrt{2}\)   (c) −8  (d) \(\displaystyle -\frac{4}{7}\) (e) −12.5 (f) -pq 2. Determine the absolute value in each of the following: (a) \(\displaystyle \frac{4}{3}-\frac{3}{2}\)     (b) −3.68 + 3.89   (c) \(\displaystyle \frac{3}{2}-\frac{4}{3}\)     (d) \(\displaystyle -\frac{1}{3}\times11+\frac{10}{3}\) (e) 0.111 − 0.889           (f) −0.111 − (−0.01001)                                                                 3. Evaluate each of the following and arrange the answers in ascending order. \(\displaystyle -\left| -4 \right|\),  \(\displaystyle \left| -12 \right|\),  \(\displaystyle \left| -5 \right|\),\(\displaystyle \left| -9 \right|\), \(\displaystyle \left| -37 \right|\),\(\displaystyle \left| 10 \right|\) ,\(\displaystyle -\left| 12 \right|\) ,  \(\displaystyle -\left| 2 \right|\). 4. Evaluate each of the following and arrange the answers in descending order. \(\displaystyle -\left| -34 \right|\), \(\displaystyle \left| 23 \right|\), \(\displaystyle \left| 29 \right|\), \(\displaystyle \left| -80 \right|\), \(\displaystyle \left| -17 \right|\), \(\displaystyle \left| 25 \right|\). 5. Evaluate each of the following expressions.  (a) -3-\(\displaystyle \left| -3 \right|\)   (b) \(\displaystyle \left| -5 \right|-\left| -3 \right|-\left| -4 \right|\)  (c) \(\displaystyle \left| -3 \right| \)+4(-2)-6     (d) \(\displaystyle \left| 4-10 \right|\) (e) \(\displaystyle -\left| -4(6-3(7-10)) \right|\)                                       6. In a certain country, the temperature on the certain day was 30° C and on the following day the temperature was 12° C use the concept of absolute value to determine the temperature change in these two days. 7. If x is an integer, represent the absolute values of the solutions to each of the following on a number line.    (a) \(\displaystyle -\frac{5}{2}\lt x\lt \frac{1}{2}\)    (b) \(\displaystyle -\frac{1}{2}\lt x\lt \frac{7}{2}\)      (c) \(\displaystyle -4\lt x\lt \frac{2}{3}\) (d) \(\displaystyle -\frac{10}{3}\lt x\lt \frac{5}{2}\)   (e) \(\displaystyle 2\le x\lt 5\frac{1}{4}\)  (f) -3 ≤ x< 3           8. The mean income of a certain company is 77 million Tanzanian shillings per year. If the income varies by 1.2 million Tanzanian shillings, what are the minimum and maximum income of the company? 9. An ice cream machine fills 160 g of ice cream in a cone. If the mass of ice cream varies by 5 g, find the heaviest and lightest cone the machine can fill. 10. In a part time job, one person target to collect Tshs 105,000 per month. Last month she was within Tshs 25,000 of her target. What is the minimum amount that she might have made last month?

Quiz

Answered0/5

Answer all questions.

1.

What will be the value of the expression (−7) + 3 × 4?

1. A) 13
2. B) 1
3. C) 5
4. D) 7

Chapter summary

1. A rational number is any number written in the form of \(\displaystyle \frac{a}{b}\), where both a and b are integers and b\(\displaystyle \neq \)0. 2. All terminating and repeating decimals are rational numbers. 3. A decimal number which neither terminates nor repeats is an irrational number. 4. An irrational number cannot be written in the form of \(\displaystyle \frac{a}{b}\), where a and b are integers and b\(\displaystyle \neq \)0. 5. Rational and irrational numbers together form the set of real numbers. 6. For any two real numbers a and b, only one of the following is true; Either a=b or a<b or a>b. 7. Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbols. 8. The absolute value of a number is its positive value, regardless of its sign.

Revision exercise

1. Write each of the following rational numbers in the form of \(\displaystyle \frac{a}{b}\), where a and b are integers and b\(\displaystyle \neq \)0. (a) \(\displaystyle 6\frac{1}{2}\) (b) \(\displaystyle -9\frac{3}{11}\)  (c) \(\displaystyle 0.\dot{8}\dot{1} \)  (d) 193.5  (e) 23  (f) \(\displaystyle 245\frac{2}{9}\)  (g) −4.3  (h)  \(\displaystyle 114.9\dot{7} \)     (i) 0.012  (j) \(\displaystyle 0.\dot{1}2\dot{3} \) 2. List twelve rational numbers which lie between: (a) −1 and 0 (b) 4 and 5 (c) −3 and −4 3. Represent on a number line all real numbers x such that x<\(\displaystyle \frac{5}{3}\). 4. Represent -4 ≤ x < 4 on a number line, where x is a real number. 5. Write an inequality for real numbers x represented on the following number line. 6. Find the absolute value in each of the following expressions.  (a) \(\displaystyle \frac{3}{8}-\frac{1}{16} + \frac{5}{4} \)                       (b) \(\displaystyle 1\frac{2}{3} +2\frac{3}{4} + 3\frac{4}{5} \)                                   (c) \(\displaystyle -\frac{17}{2}+\frac{1}{10}-3\frac{2}{7}\)     (d) 62.15 -88.765+45.678   (e) \(\displaystyle 2\frac{1}{3}+4\frac{1}{5} + \frac{1}{2} \)       (f) \(\displaystyle \left( 7\frac{5}{8}+3\frac{1}{4} \right) - \left( 2\frac{2}{5}+4\frac{1}{2} \right) \) 7. Find the absolute value in each of the following expressions.   (a) \(\displaystyle \frac{2}{3}\times3\frac{9}{16}\)   (b) \(\displaystyle 2\frac{1}{2}\times\frac{7}{8}\div\left( -3\frac{3}{4} \right)\)   (c) \(\displaystyle -\frac{4}{5}\div \frac{33}{25}\)  (d) \(\displaystyle 6\frac{1}{4}+\left( 4\frac{1}{2}\div 2 \right)\)   (e) \(\displaystyle -\frac{4}{5}\times\left( \frac{3}{7}+4\frac{7}{10} \right)\)          (f) \(\displaystyle 5\frac{1}{2}-\left( -7+\frac{3}{11} \right) \)                           (g)\(\displaystyle \frac{\frac{1}{2}+\frac{1}{3}}{\frac{2}{3}+\frac{4}{5}+1\frac{1}{4}}\)   (h) \(\displaystyle \frac{6\frac{2}{3}\times2\frac{4}{5}-3}{\left( 2\frac{2}{3}\times 2\frac{4}{5} \right)-3}\)    (i) \(\displaystyle -\frac{16}{5}\times \left( -\frac{3}{8}\times -\frac{2}{9} \right) \) 8. Write the following using the recurring decimal notation, hence convert each of the decimals into fractions. (a)0.22442244… (b)0.123123123123… (c)1.578578578… (d)0.1010101010…(e)2.153846153846…                       9. Write each of the following fractions into decimals. (a) \(\displaystyle \frac{5}{9} \)  (b) \(\displaystyle \frac{14}{11} \) (c) \(\displaystyle \frac{29}{6} \) (d) \(\displaystyle \frac{2}{27} \) (e) \(\displaystyle \frac{5}{12} \)   (f) \(\displaystyle \frac{5}{11} \)     (g) \(\displaystyle \frac{40}{33} \) (h) \(\displaystyle \frac{4}{27} \) (i) \(\displaystyle \frac{7}{15} \) (j) \(\displaystyle \frac{1}{13} \)                                                                                                             10. If x = \(\displaystyle 0.\dot{1}\dot{5} \)  and y = \(\displaystyle 0.\dot{3} \), find the value in each of the following in fraction. (a)  \( x + y \)      (b) \(\displaystyle \frac{x}{y} \)         (c) \(\displaystyle \frac{y}{x} \) 11. Identify each of the following as rational or irrational numbers. 0.75,  \(\displaystyle\frac{90}{12007} \),  \(\displaystyle\sqrt{7} \),  12,  \(\displaystyle \frac{1}{\sqrt{2}} \) 12. Identify whether or not, a mixed number\(\displaystyle -3\frac{1}{2} \) is rational. 13. Write all real numbers from 5 to 6. 14. A rational number N is equal to \(\displaystyle \frac{2}{5} \) times the sum of \(\displaystyle \frac{24}{7} \) and \(\displaystyle \frac{1}{14} \). Find the number N. 15. Identify fractions which are equivalent to each of the following decimals. (a) 2.344444444… (b) 21.305305305305… (c) 0.17171717… 16. If x = \(\displaystyle 1.\dot{3} \)  and y= \(\displaystyle 0.\dot{2}\dot{1} \), find the value of xy and x - y. 17. The average number of tomatoes in a heap is 28. The number of tomatoes vary by three. What is the maximum number of tomatoes that could be in a heap. 18. The mean distance from Morogoro to Chalinze is 90 km. However, due to errors in measurements, it varies by 1.6 km from different travel data companies. What is the possible maximum and minimum distance from Morogoro to Chalinze? 19. Runa bought a land for Tshs 2,562,000 and made a profit of Tshs 1,225,600 after selling it. How much did he sell the land? 20. How many years are in 5,840 days if a year has 365 days? 21. If a pond has a capacity of 35,000 litres of water, how long does it take to be filled by a hose with a capacity of filling 13 litres per minute? 22. Write down the inequalities which correspond to each of the following number lines.       23. Represent each of the following information on a number line.  (a) The number of football players allowed on the pitch for match is not less than 8 and does not exceed 11.  (b) Maximum volume of a container is at least 1.5 litres  (c) In a class, the tallest student is 160 cm and the shortest is 100 cm.  (d) The lowest temperature in April was 4° C. 24. If x= \(\displaystyle 0.\dot{2} \), y= \(\displaystyle 0.\dot{3} \),  and z= \(\displaystyle 0.\dot{9} \),  evaluate each of the following expressions. (a) xy      (b) x+yz      (c) \(\displaystyle \frac{xz}{y} \)+y      (d) xy+xz+yz 25. Evaluate each of the following expressions. (a) \(\displaystyle 1.02\dot{5}-0.80\dot5 \)   (b) \(\displaystyle 0.0\dot{6}-0.0\dot5 \)      (c) \(\displaystyle 0.\dot{6}\dot8-0.0\dot3\dot1 \)      (d) \(\displaystyle 0.\dot{4}\div 0.\dot2 \)

Project: Appreciating numbers in daily life.

Write an essay that describe with real-life examples on how numbers are used in various real-life activities.