## How to convert from decimal to other number systems

This post will be of special interest for people who are regularly in touch with mathematics. Students preparing for competitive examinations usually have Base System (Number Systems) in the list of their topics under quantitative aptitude.

Conversion from decimal to binary and other number bases

In order to convert a decimal number into its representation in a different number base, we have to be able to express the number in terms of powers of the other base. For example, if we wish to convert the decimal number 100 to base 4, we must figure out how to express 100 as the sum of powers of 4.

100 = (1 x 64) + (2 x 16) + (1 x 4) + (0 x 1)

= (1 x 4^3) + (2 x 4^2) + (1 x 4^1) + (0 x 4^0)

Then we use the coefficients of the powers of 4 to form the number as represented in base 4: Read More

## Comparison of Fractions

In this post you will learn about comparing fractions and arranging them in ascending or descending order of magnitude.  All of us must have learned this during our school days. However, here we are going to discuss this in more detail and will discover the most suitable way of doing it.

Fractions can be compared in many ways. Here we’ll discuss 4 different ways of doing it.

## If Denominators are same and Numerators are different

Like

56              57

—    and  —

98              98

Just compare the numerators,

So the rule is:

a         b

— > —,       if  a > b

n         n

Since 56 < 57,

56        57

—    < —

98         98

They are in the same order as their numerators.

## If Numerators are same and Denominators are different

Compare 5/9 and 5/6

This works the opposite way: The bigger the denominator, the smaller the fraction. So the rule

is:

n        n

— > —  if  a < b

a         b

Since 9 > 6, 5/9 < 5/6. They are in the reverse order of their denominators.

## General Cases

You simply convert the fractions to the first case, by giving them a common denominator.

You do not really have to worry about finding the least common denominator, though sometimes that will save a lot of work.

Let us compare 5/9 and 4/7. Since we do not see any common factors immediately (and in fact there are not any), we can just multiply the denominators to get a common denominator, 63. To convert 5/9 to 63rds, we multiply by 7; to convert 4/7 to 63rds, we multiply by 9:

5              4

— ,       —

9             7

5*7          4*9

—         ?     —

9*7           7*9

35 < 36,

so

5            4

— <     —

9             7

You may not calculate the value of denominator because it will be same in both cases.

5*7=35          4*9=36; since this is bigger, 4/7 is bigger

If there is a common multiple in denominators, for instance, which is bigger, 5/9 or 44/81? I see that

81 is a multiple of 9, so I do not have to go to the trouble of multiplying 5 by 81 and 44 by 9; I just multiply 5 by 9 and compare to 44:

5         44

— ,     —

9          81

5*9       44

—     ,   —

9*9       81

45 > 44 so

5           44

— >       —

9            81

## Converting Fractions into Decimal Form

Here we need to convert each one of the given fractions in the decimal form. Thereafter, arrange them in ascending or descending order