Subsidiary Mathematics

My goals

Introduction

Real numbers are all Rational and Irrational numbers. They can also be positive, negative or zero. A simple way to think about the Real Numbers is: any point anywhere on the number line. Powers (convenient way of writing multiplications that have many repeated terms), radicals (an expression that has a square root, cube root, …), a logarithm (the power to which a number must be raised in order to get some other number) of a number .... are all real numbers. 

If you have bacteria that divide every 40 minutes and are currently taking up 0.1% of the petri dish, you can use logarithms to estimates how long it will take them to fill up the entire dish. The same goes for 2000 francs in an account, logarithms will tell you when you will have 4000 francs. In finance and business logarithms can be useful for calculating compound interest. 

Required outcomes 

After completing this unit, the learners should be able to:

1. Absolute value and its properties

Activity 1

Draw a number line and state the number of units that are between; 

Absolute value of a number is the distance of that number from the original (zero point) on a number line. The symbol 

||  is used to denote the absolute value.

Example 1

7 is at 7 units from zero, thus the absolute value of 7 is 7 or |7| = 7. Also -7 is at 7 units from zero, thus the absolute value 

Solution

Solution

Exercise 1 

Find the value(s) of x 

Activity 2

Evaluate the following operations and compare your results in eache case 

Properties of the Absolute Value 

Opposite numbers have equal absolute value. 

Example 4

The absolute value of a product is equal to the product of the absolute values of the factors. 

Example 5

The absolute value of a sum is less than or equal to the sum of the absolute values of the addends. 

Example 6

Exercise 2 

Simplify: 

2. Powers and radicals

Powers in IR

Activity 3

Peter suggested that his allowance be changed. He wanted $2 the first week, with his allowance to be doubled each week. His parent investigated the suggestion using this table

1. Complete the table to find how many dollars Peter would be paid each of the first five weeks. 

2. How much would Peter be paid the seventh week? The tenth week? 

3. Do you think his parent will agree with his suggestion? Explain.

Example 7 

Notice

Properties of powers 

These properties help us to simplify some powers. There is no general way to simplify the sum of powers, even when the powers have the same base. For instance,  is not an integer power of 2. But some products or ratios of powers can be simplified using repeated multiplication model of n a.

Example 8 

Exercise 3 

Simplify

Radicals in real numbers

Activity 4

Evaluate the following powers 

Example 9 

Example 10

Properties of radicals

Example 11

Simplify:

Solution

Exercise 4

Simplify:

Operations on radicals 

When adding or subtracting the radicals, we may need to simplify if we have similar radicals. Similar radicals are the radicals with the same indices and same bases. 

Activity 5

Simplify the following and keep the answer in radical sign

Addition and subtraction 

When adding or subtracting the radicals we may need to simplify if we have similar radicals. Similar radicals are the radicals with the same indices and same bases.

Example 12

Exercise 5

Simplify

Rationalizing

Activity 6

Make the denominator of each of the following rational;

Rationalizing is to convert a fraction with an irrational denominator to a fraction with rational denominator. To do this, if the denominator involves radicals, we multiply the numerator and denominator by the conjugate of the denominator. 

Example 13

Exercise 6

Rationalize the denominator; 

3. Decimal logarithms and properties

Activity 7

What is the real number for which 10 must be raised to obtain 

    1. 1                          2. 10                    

    3. 100                      4. 1000                    

    5. 10000                  6. 100000

The decimal logarithm of a positive real number x is defined to be a real number y for which 10 must be raised to obtain x. 

Example 14

log (100) = ? 

We are required to find the power to which 10 must be raised to obtain 100

So log(100) = 2

y= log x means 10^y = x 

Be Careful! 

Example 15

Example 16

Example 17

Solution

Example 18

Solution 

Co-logarithm 

Co-logarithm, sometimes shortened to colog, of a number is the logarithm of the reciprocal of that number, equal to the negative of the logarithm of the number itself, 

Example 19

Exercise 7 

 

Unit summary