Topic 4: Numerical Concepts 1 (Indices)

By the end of this topic, you will be able to:

(a) give approximate answers to calculations.
(b) write numbers to a given number of significant figures.
(c) differentiate between significant figures and decimal places.
(d) express numbers in standard form.

(e) identify base number and index.

(f) state and apply the laws of indices in calculations.
(g) use a calculator to find powers and roots.

Keywords

• decimal place
• index
• powers
• roots
• significant figures
• standard form

Introduction

Your father has land measuring 235.6 m and 47.2 m. How would you let your father know the area roughly?
In this topic, you will learn numerical concepts and how to give answers to the calculations using certain criteria such as rough estimate based on significant figures or decimal places. You will also understand and use indices and standard form.

4.1 Giving approximate answers to calculations

In this section, you will understand the meaning of approximation and how useful it is in computation.

Activity 4.1 Getting familiar with approximation
Look at the picture of a chalkboard provided below. Use it to answer the questions that follow.

1. What do you think is the area of any object around you?
2. Explain how you have obtained the area in (1).
3. What possible instruments would you use to calculate the area of the chalkboard?
4. Give reasons for the choice of instruments you have selected in (3).
5. Use appropriate instruments to find the accurate area of the classroom chalkboard.
6. Compare the area you got in (1) to the area you got in (5) and comment on them.
7. Share your findings with the rest of the class.

Learning points

The area of the chalkboard in (1) is the estimated or approximated value or guessed value while the area obtained by measurement is the exact value.
The difference between the two areas should be very small (negligible).

Quiz

Choose TRUE or FALSE for the following.

1. The height of a baby is 10 m.
2. The area of Uganda is approximately 241,037 km2.
3. The average age of your class is 6 years old.
4. An exercise book weighs approximately 10 kgs.

Activity 4.2 Computing approximate values
Senior Two learners belong to a mathematics club. One of the club activities is to conduct quiz. Four learners competed in a 30-second task in which they were required to perform a computation of 26×43.
The learners did it as follows:

1. Who of the four learners took a shorter time to get the approximate answer?
2. What can you comment about learner B?
3. Comment about the answers for learner C and learner D in relation to the exact answer.
4. Comment about learners A in terms of approximation.

Exercise 4.1
Choose the most correct answer for questions 1 to 6.

7. A plot of land has a length of 28.7 m and a width 17.3 m. A farmer is interested in buying the land. Unfortunately, the farmer does not have a calculator, so the farmer will have to mentally estimate the area of the land. What is the best estimated area for the plot of land that the farmer will buy?

8. An engineer requires 212 litres of diesel to run a generator for a day. The price of diesel at the petrol station is UGX 3,495. Estimate how much money the engineer will need to buy the diesel required.

9. A shopkeeper is filling some jars with various sweets. Each jar can hold 278 sweets. If there are only 27 jars, estimate the total number of sweets the shopkeeper has.

4.2 Writing numbers to a given number of significant figures

In this section, you will understand what a significant figure is and its importance in computation.

Activity 4.3 Writing numbers in significant figures
A Learner went to a tailor to sew a new long-sleeved shirt for his uniform.
The tailor took measurements of the learner as follows.

The tailor is meant to buy material for the entire long-sleeved
shirt.

1. What is the total material for all the different parts of the shirt that the tailor measured?
2. How much material will the tailor need to buy?
3. Give reasons why the tailor bought the amount of material you obtained in (2).

1. For each of the parts of the long-sleeved shirt:
   • (a) Write down the number of digits in each measurement.
   • (b) Write the significant figures in each of the measurements.
   • (c) What can you deduce about the zero in the measurements of the front panel, sleeve, collar band, collar and pocket?
   • (d) Does the zero in each of these parts mean the same as a zero between 8 and 5 for 0.0805? Give reasons for your answer.

Learning points
1. Significant figures are the number of digits in a value, often a measurement, that contribute to the degree of accuracy of the
value.
2. You start counting significant figures at the first non-zero digit

Quiz
1. Identify the number of significant digits/figures in the following given numbers.
(a) 45
(b) 0.046
(c) 7.4220
(d) 5002
(e)3800

2. Solve 4.76 + 5.62 + 33.21 and state the number of significant digits/figures.
3. What is the number of significant digits from the following computations?
5.2 x 103 x 6.732 x 103

Rounding significant figures

When the first digit in left is less than 5, the last digit held should remain constant. When the first digit is greater than 5, the last digit is rounded up. When the digit left is exactly 5, the number held is rounded up or down to receive an even number. When more than one digit is left, rounding off should be done as a whole instead of one digit at a time.

There are two rules to round off the significant numbers:
1. First, we have to check up to which digit the rounding off should be performed. If the number after the rounding off digit is less than 5, then we have to exclude all the numbers present on the right side.

2. But if the digit next to the rounding off digit is greater than 5, then we have to add 1 to the rounding-offExercise 4.2

1. Write 45.378212 correct to 3 significant digits/figures. Round off 0.0012485 m correct up to 3 significant figures. 3. Round off:
(a) 15.256 g correct to the three significant figures.

(b) 0.00838 mg correct to the two significant figures.

(c) 0.004509 g correct to the one significant figure.

(d) 29.404 mm correct to the four significant figures.

(e) 22.311 km correct to the four significant figures.

(f) 3.0921 kg correct to the two significant figures.

(g) 0.003519 mm correct to the one significant figure.

2. Find the area of a rectangle BCDE whose measurements are 26.43 metres by 35.89 metres, giving your answer to 5 significant figures.

Examples

1. 8726 rounds up to 9000, because the 2nd significant digit is a 7.
2. 73.283 rounds down to 70 because the 2nd significant digit is a 3.
3. 152 rounds up to 200 because the 2nd significant digit is a 5.
4. 0.003826 rounds up to 0.004 because the 2nd significant digit is an 8.
5. 0.60828 rounds down to 0.6 because the 2nd significant digit is a 0.

Exercise 4.2

1. Write 45.378212 correct to 3 significant digits/figures.
2. Round off 0.0012485 m correct up to 3 significant figures.
3. Round off:
   • (a) 15.256 g correct to the three significant figures.
   • (b) 0.00838 mg correct to the two significant figures.
   • (c) 0.004509 g correct to the one significant figure.
   • (d) 29.404 mm correct to the four significant figures.
   • (e) 22.311 km correct to the four significant figures.
   • (f) 3.0921 kg correct to the two significant figures.
   • (g) 0.003519 mm correct to the one significant figure.
4. Find the area of a rectangle BCDE whose measurements are 26.43 metres by 35.89 metres, giving your answer to 5 significant figures.

4.3 Writing numbers to a given number of decimal places

In this section, you will understand what a decimal place is and its importance in computation. You will be in a position to differentiate between a decimal place and significant figure.

Activity 4.4 Distinguishing between decimal places and significant figures
A learner went to a tailor to sew a new long-sleeved shirt for his
uniform.
The tailor took measurements of the learner as follows.

1. Write down the number of digits in each measurement.
2. Write the decimal places in each of the measurements.
3. What can you deduce about the zero in the measurements of: the collar band? (b) the collar?
4. How many decimal places and significant numbers are in 0.08?

Activity 4.5 Comparing decimal places and significant figures
1. Complete the table by rounding the following numbers as
stated.

2. Compare the results of 1 decimal place with 1 significant figure. What can you deduce about them?
3. Compare the results of 2 decimal places with 2 significant figures. What can you deduce about them?

Exercise 4.3

1. Find the circumference of a circle whose radius is 8.24 cm correct your answer to:
   (a) 4 places of decimal.
   (b) 4 significant figures.
2. Find the area of the circle whose radius is 8.24 cm correct to:
   (a) 3 significant figures.
   (b) 3 places of decimal.
3. Find the value of:
   (a) 23.37 correct to 1 decimal place. (b) 7.755 correct to 2 decimal places. (c) 9.99 correct to 1 decimal place.
   (d) 101.2345 correct to 3 decimal places. (e) 121.513 correct to 2 decimal places.
4. What is the nearest whole number of 13.607
5. What number has the place value to the nearest tenths of 3.105?
6. What number has the place value to the nearest hundredths of 41.252
7. What number has the place value to the nearest thousandths of 56.3009
8. Round off to the nearest whole number of 9.57.

 4.4 Identifying the base and index of a number

Activity 4.6 Writing numbers in index form
Take a long strip of paper 60 cm long and 10 cm wide. Fold it in two equal parts, then fold the two equal parts into two equal parts. Do it successively 6 times. Record your results as follows:

Exercise 4.4
1. In the expression 2″ what name is given to:
(a) 2?
(b) n?
2. Write in index notation: 2 x 3x3x5x5x5x2.

3. What is 45 in expanded form?
4. What is a3 x b2 in expanded form?

5. Which of the following has a base number and an index? Write the possible ones in the form of indices.
(a) 8
(b) 67
(c) 32/51
(d)12
(e) 81
6. What is the value of 3 x 42 x 32?
7.Using 3 as the base, write each of the following in index form. (a) 81
(b) 729
8. Using 5 as base, write the following as indices.

(a) 1
(b) 5
(c)625
9. (a) Write 450 as the product of its prime factors.
(b) Write your answer in index form.
10. Write X 2/3*2/3*3/3*2/3 in index form.

4.5 Stating and applying the laws of indices in calculations

In this section, you will learn how to derive the different laws of indices and how they are useful in computation. In primary school, you learnt about powers.

Activity 4.7 Using laws of indices
Learners were given the following product to perform 3 x 33. Different learners answered it as follows.

3. A square field has an area of 25 m2. How long is each side?
4. A cube-shaped fish tank has a volume of 216 cm3. How long is each side?

Negative and zero indices

You should be pretty confident with simple numerical indices by now. Negative indices are all powers that have a minus sign in front of them and are, as a result, negative. They are quite easy to deal with as there is only one thing that you have to do. Just quickly have a look at the following examples.

4.7 Expressing numbers in standard form
In this section, you are going to learn that standard form is a convenient way of writing very large or very small numbers. It is used on a scientific calculator when a number is too large or too small to be displayed on the screen.

4.7 Expressing numbers in standard form

In this section, you are going to learn that standard form is a convenient way of writing very large or very small numbers. It is used on a scientific calculator when a number is too large or too small to be displayed on the screen.

Activity 4.9 Expressing numbers in standard form
On 31 March 2020, the government sought a further 284 billion Ugandan shillings (UGX) to cater for the government’s response to the Covid-19 pandemic. This was broken down as detailed in Table 1. Table 1

1. In Table 1, the approved amount of money for health was UGX104,188,234,110. Olwech, Natukunda and Ismail expressed the amount of money as a base of 10. Each one of them was writing the large number in a simpler way as follows:

(a) Which one of them expressed the number in the smallest way? Give reasons for your answer.
(b) Use your calculator and feed in the amount 104,188,234,110 and press the enter key. What do you observe on the screen of your calculator? Discuss with your friend the notation in which the amount of money has been expressed.

(c) Compare the outcome on the screen of the calculator in (b) with the different ways in which Olwech, Natukunda and Ismail expressed the amount of money. What do you learn from the finding?

2. Express all the amount of approved money in the remaining sectors in the smallest way possible as a base of 10.
3. Share your work with the rest of your classmates.

Learning points
A large number can be expressed as a base of 10, with the decimal point placed immediately after the first non-zero digit from the left. Notice that the power raised to the base of ten is the number of digits counted from the right to the position of the decimal point.
The large numbers have the power as a positive. This process explains what standard form means.
Standard form is sometimes referred to as scientific notation.

Quiz

Fill in the correct response.
1. 27,000 written in standard form is ………..
2. 1.5 x 105=………….
3. Write twenty thousand, nineteen in standard form.
4. Choose the most correct option. Rewrite in standard form 23.5 x 104.
(a) It is already in standard form
(b) 2.35 x 103

(c) 2.35 x 105

(a) What is the equivalent weight in kilograms for the proteins?
(b) Express your answer in standard form.
(c) Express 1.1 grams of protein in standard form.
(d) How would you describe 0.11 and 1.1 in terms of size (large or small) numbers? What effect would the size of numbers have on the powers to a base of 10?

(e) As shown in the table, fat takes a quantity of 0.3 grams. Express this quantity in standard form.
(f) Write all the other quantities in standard form.

2. Share your work with the rest of your classmates

Learning Points

A small number can be expressed as a base of 10, with the decimal point immediately after the first non-zero digit from the left Notice that the power raised to the base of ten is the number of digits counted from the right of the decimal point to the first non zero digit

Small number have the power as a negative. This process explains, what standard form means.

Standard form Is sometimes referred to as scientific notation.

5. The distance from Kampala to Entebbe through the expressway is about 44,000 metres. Write this distance in standard form.

6. In an election, one candidate obtained (5.3452 x 10″) votes and another candidate obtained (7.6548 x 10″) votes. If there were (2.5698 x 10″) voters who stayed at home and failed to vote, find the number of voters who had registered in that constituency.

7. Two candidates were contesting for mayor and the constituency had (9.38426 x 10″) votes. If the candidate who lost obtained (8.4623 x 10″) votes, find the votes which the winner obtained, assuming that everybody in the constituency had registered and voted.

8. The distance from Kampala to Jinja is estimated to be 95,000 metres by expressway. Write this distance in standard form.

9. The diameter of a small tiny hole in a vessel is 0.000528 cm. Write this figure in standard form.

10. Sir Isaac Newton (1642-1727) was a mathematician, physicist and astronomer. In his work on the gravitational force between two bodies, he found that he needed to multiply their masses together.
(a) Work out the value of the mass of the earth multiplied by the mass of the moon. Give your answer in standard form.

Mass of earth = 5.98 x 1024 kg
Mass of moon = 7.35 x 1022 kg

(b) Work out the square of the distance between the earth and the moon. Give your answer in standard form.
Distance between earth and moon = 3.89 x 105 km

4.7 Using a calculator to find powers and roots of numbers

Provided you have a scientific calculator.
Consider symbols such as x to write an index where x is the base and is the index, then press = to solve. Take note of symbols such as (y) to find roots of numbers where x is the index and y is the base.

5. Philimeon is trying to complete his maths homework, but cannot remember the rule to simplify (23)4. Belinda says the rule is multiply the powers so the answer is 212; and Abubakar says the rule is add the powers so the answer is 27. Produce a step-by-step explanation to convince the three friends of the correct answer.