Mathematics

Thinking Mathematically

7th Edition

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Thinking Mathematically – eText (w/Access Code – MyMathLab ) – ISBN # 9780134705095

Number Representation and Calculation

As you are going through the sections in Chapter 4, you will notice that we are working with different bases.

Please explain to the class how to:
Rewrite a number in a base other than ten
Add two or three numbers in a base other than ten
Subtract two numbers in a base other than ten
Why do we learn to work with numbers in different bases?
Please provide detailed explanation and numerical examples for each answer.

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ANSWER

Number representation and calculation

1. Rewriting a number in a base other than 10

Steps involved are:

1. Divide the number by the base using long division. The remainder becomes the first least significant digit of the resulting number in the new base.
2. Divide the quotient of the above step by the new base. The remainder is the second least significant.
3. In a similar manner repeat the process until the quotient is less than the base. The quotient which is the last digit becomes the most significant digit.
4. Read the remainders from the bottom to the top
5. Write the answer and include the new base as the subscript

Example is as shown below:

Write 55 to base 8

Answer is 67₈

1. Adding numbers in the same base other than 10

Base 10 is the most common used.

It has 10 digits which are: 0       1          2          3          4          5          6          7          8          9

A number line for the base ten will be written as

0  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15….

The same criteria is used for other bases. For instance, digits for base 3 are 0       1          2

The number line is

0  1  2  10  11  12  20     21    22   100  101  102 110 111 112 …

Adding two numbers in base 3

1201₃ + 122₃

=  1 2 0 1₃

+ 1 2 2₃

2 1 0 0₃

• Subtracting numbers in a base other than 10

When subtracting in base 10, it is common to borrow from the next digit when you are subtracting a larger digit from a smaller digit. The borrowed value is added to the smaller digit as a base 10. In a similar manner, a borrowed value is added in terms of the base.

For instance, borrowing a value in  a number of base 5. The borrowed value is added to the smaller digit as 5 since the number is to base 5.

Example: subtract 412₅ – 213₅

= 4 1 2₅

-2 1 3₅

2 is smaller than 3, therefore borrow 1 from next digit. Add the borrowed value as 3 to the digit 2. This becomes

= 4 0 5₅

-2 1 3₅

2

In a similar manner, 0 is less than 1, thus borrow 1 from 4 and add the borrowed value to 0 as 3. This becomes

= 3 3 5₅

-2 1 3₅

2 2

3 subtract 2 is 1. Thus the final result is

= 4 1 2₅

-2 1 3₅

1 2 2

1. Learning to work with numbers in different bases

This is important because it helps in the understanding that base 10, which is commonly used, is not the only base that can be used. Thus, numbers can be expressed in many bases other than 10. As such, the concept of number bases and place values among others is better understood.

Knowledge of other number bases is important because these are used in other areas such as computers which use binary and hexadecimal numbers.