Systems of Numeration

We know two systems of numeration.

(i) Hindu-Arab System of numbers based on 10 digits, i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

(ii) Roman System of numbers based on 5 digits, i.e., I, V, X, L, C, D and M.

The numbers based on 10 digits and 5 digits may be interchanged.

(iii) There is a third system of numbers named as computer system. This system of numbers is based on two digits, i.e., 0 and 1.

This system is also called 2-digit based number system.

The numerals of numbers in the three systems are as follows.

10-digit-based numbers: 358293, 934528

5-digit-based numbers: XL, CDXXVI, MMC

2-digit-based numbers: 1101011, 1101111

The two-digit based numbers and 10-digit based numbers may be interchanged.

In 10-digit based numbers, the place values from right to left are as follows:

Place Values


In 2-digit based numbers, the place values from right to left are given below.

Place Values Chart

Let there be a 2-based number (1101011)2 and we have to write it as a 10-based number:

The 2-based number is written in its extended form (i.e., according to the place value) as shown here:

Place Value Number

Therefore, Number = 64 + 32 + 0 + 8 + 0 + 2 + 1 = 107= (2-based number) 1101011 = (10-based number) 107

In short 2-based number 1101011 may be changed into 10-based number



Therefore, Number = 64 + 32 + 0 + 8 + 0 + 2 + 1 = 107 = (2-based number) 1101011 = (10-based number) 107

In short 2-based number 1101011 may be changed into 10-based number

(1101011)2 = 1 x 26 + 1 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20

          = (1 x 64) + (1 x 32) + (0 x 16) + (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1)

          = 64 + 32 + 0 + 8 + 0 + 2 + 1

          = (107)10

The 10-based number may also be changed into 2-based number.



Say, we have to change (107)10 into 2-based number.

10-based Numbers


one’s place

2’s place

4’s place

8’s place

16’s place

32’s place

64’s place



2-based number = 11010111

(i) 107 is divided by 2, quotient is 53 the remainder is 1.
This remainder 1 is the digit of 2-based number having place 20 = 1

(ii) 53 is divided by 2, quotient is 26 the remainder is 1, it is the digit of 2-based number having place value 21 = 2

(iii) 26 is divided by 2, quotient = 13, R = 0.
Remainder 0 has the place value 22 = 4 where 0 x 4 = 0

(iv) 13 is divided by 2, quotient = 6, R = 1, place value = 23 = 8, 1 x 8 = 8

(v) 6 is divided by 2, Quotient = 3, R = 0, place value of 0 = 24, 0 x 16 = 0

(vi) 3 is divided by 2, quotient = 1, R = 1, place value of 1 x 25 = 1 x 32 = 32

(vii) 1 is divided by 2, quotient = 0, R = 1,place value = 1 x 26 = 1 x 64 = 64

Therefore, (107)10 = (1101011)2

2-based number = 1101011

10-based number = 64 + 32 + 0 + 8 + 0 + 2 + 1 = (107)10

Say we have to change (119)10 into 2-based number.

2-based number


20 = 1-place

21 = 2-place

22 = 4-place

23 = 0-place

24 = 16-place

25 = 32-place

26 = 64-place



Verification

    1     1     1     0     1     1     1

= 26 + 25 + 24 + 23 + 22 + 21 + 20

= 64 + 32 + 16 + 0 + 4 + 2 + 1

= 119

2-based number

= 1110111

10-based number

= 64 + 32 + 16 + 0 + 4 + 2 + 1

= 119

Related Concept

Patterns and Mental Mathematics

Counting Numbers in Proper Pattern

Odd Numbers Patterns

Three Consecutive Numbers

Number Formed by Any Power

Product of The Number

Magic Square

Square of a Number

Difference of The Squares

Multiplied by Itself

Puzzle

Patterns









4th Grade Math Activities

From Systems of Numeration to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Worksheet on Word Problems on Fractions | Fraction Word Problems | Ans

    Jul 16, 24 02:20 AM

    In worksheet on word problems on fractions we will solve different types of word problems on multiplication of fractions, word problems on division of fractions etc... 1. How many one-fifths

    Read More

  2. Word Problems on Fraction | Math Fraction Word Problems |Fraction Math

    Jul 16, 24 01:36 AM

    In word problems on fraction we will solve different types of problems on multiplication of fractional numbers and division of fractional numbers.

    Read More

  3. Worksheet on Add and Subtract Fractions | Word Problems | Fractions

    Jul 16, 24 12:17 AM

    Worksheet on Add and Subtract Fractions
    Recall the topic carefully and practice the questions given in the math worksheet on add and subtract fractions. The question mainly covers addition with the help of a fraction number line, subtractio…

    Read More

  4. Comparison of Like Fractions | Comparing Fractions | Like Fractions

    Jul 15, 24 03:22 PM

    Comparison of Like Fractions
    Any two like fractions can be compared by comparing their numerators. The fraction with larger numerator is greater than the fraction with smaller numerator, for example \(\frac{7}{13}\) > \(\frac{2…

    Read More

  5. Worksheet on Reducing Fraction | Simplifying Fractions | Lowest Form

    Jul 15, 24 03:17 PM

    Worksheet on Reducing Fraction
    Practice the questions given in the math worksheet on reducing fraction to the lowest terms by using division. Fractional numbers are given in the questions to reduce to its lowest term.

    Read More