# Numeral system

It is a writing system for expressing numerals, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a standard manner. It also reflect  the algebraic and arithmetic structure of the numbers (positional notation).

Number: A number is an idea about counting & measurement.

Numeral: A numeral is a symbol or word that represents a number. It is made up of digits.

π is a special number that can’t be written exactly.

Different types of numeral systems

Unary Numeral System (base-1)(tally marks)

Binary Numeral ystemS (base-2)

Decimal  Numeral System (base-10)

Sexagesimal Numeral System (base-60)

Roman Numeral System (Letters)

# Decimal Numeral System

It is most commonly used numeral system. Ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) are used to represent numerals uniquely by arranging them at different position called Places (ones, tens etc). Every next position to the left is ten time bigger than the position to its right.

## Structure of Integers

Place at the most right in a numeral is called units or ones place.

( Place value of the digit at ones =1 ×face value of the digit)

Place at the left of ones is called tens place.

( Place value of the digit at tens = 10 ×face value of the digit)

Place at the left of tens is called hundreds place.

( Place value of the digit at hundreds = 100 ×face value of the digit)

Place at the left of hundreds is called thousands place.

( Place value of the digit at thousands = 1000 ×face value of the digit)

e.g., 124 can be expanded as

124 = 100×1 + 10×2 + 1×4

Can a fraction be represented in decimal numeral system ?

Every next position to the left is ten time bigger than the position to its right. This implies that if we move from left to right every place get ten times smaller.

But what if we continue past ones? What is ten times smaller than ones?

It is 1/10 called tenths.

Place next to the right of tenths (& ten time smaller) is called hundredths (1/100).

Place next to the right of hundredths (& ten time smaller) is called thousandths (1/1000).

# Decimal numbers

Numbers having both integral part and fractional part is called decimal numbers or simply decimals.
↪ To differentiate between integral part and fractional part in a decimal, we put a point (or dot) in between called decimal point (•).

↪ e.g., 12.47 (read as twelve and forty seven hundredths or twelve point four seven)

12 on the left side of decimal point is the integral part. 1 is at tens place & 2 is at units place.

47 on the right of decimal point is the fractional part. 4 is at tenths place & 7 is at hundredths place.

↪ We can expand 12.47 as following

12.47 = 10×2 + 1×2 + 1/10 ×4 + 1/100 ×7

↪ We can write a decimal as :

→ Integer plus decimal fractions (tenths, hundredths …)

12.47 = 12 + 4/10  + 7/100

→ Integer plus decimal fraction or mixed fraction

12.47 = 12 + 47/100 = 12

→ Decimal fraction

12.47 = 1247/100

↬ Decimals enable us to write very large numbers as well as very small numbers.

We can read a decimal in two ways :

→ The Integer (of the integral part) + AND + the Decimal Fraction (of the fractional part).

We read “and” in place of  the decimal point.

45.7 – Forty five and seven tenths

21.57 – Twenty one and fifty seven hundredths

3.576 – Three and five hundred seventy six thousandths

0.25 – Twenty five hundredths

→ The Integer (of the integral part) + POINT + the individual digits (of the fractional part)

45.7 – Forty five point seven

21.57 – Twenty one point five seven

3.576 – Three point five seven six

0.25 – Zero point Two five / Point two five

## Representing Decimals on number line

→ Locate integral part on the number line.

→ Divide the next unit distance into 10 (for decimal up to 10th place) or 100 (for decimal up to 100th place) equal parts.

→ Locate the fractional part.

→ e.g., to show 1.5 on number line, first locate the integral part one on the number line. Next we have to add .5 (or 5/10) to 1, for that we divide distance between 1 & 2 in 10 equal parts. 5th part on this division shows 5/10 .Thus, this point is 1.5.

## Decimal to fraction

↪ 0.1 = 1/10,

0.2 = 2/10 = 1/5 ( in the lowest term)

0.3 = 3/10

↪ 0.01 = 1/100,

0.02 = 2/100 = 1/50 ( in the lowest term)

0.37 = 37/100

0.75 = 75/100 = 3/4 ( in the lowest term)

3.75 = 375/100 = 15/4 ( in the lowest term)

∴ To write decimal as fraction, first we count the number of digits after decimal point then put the same number of zeros after 1 in the denominator of the fraction and then, simply the fraction to its lowest term.

## Fraction to decimal

We know that in decimal notation,

1/10 = .1,

1/100 = .01,

1/1000 = .001

Fraction with denominator 10

↬ 1/10 = .1,

2/10 = .2

So, for the proper fraction with denominator 10 (or any single digit integer n),

n/10 = .n

5/10 = .5

↬ For improper (or mixed) fraction with denominator 10

17/10 = 1 + 7/10 = 1 +.7 = 1.7

23/10 = 2 + 3/10 = 2 +.3 = 2.3

157/10 = 15 + 7/10 = 15 +.7 = 15.7

So, for the improper (or mixed) fraction with denominator 10, place decimal point before first digit from the right of the number.

Fraction with denominator 100

↬ 1/100 = .01,

2/100 = .02,

21/100 = .21

So, for the proper fraction with denominator 100

n/100 = .0n  (any single digit integer n),

m/100 = .m  (any double digit integer m),

↬ For improper (or mixed) fraction with denominator 100

= 1+ 57/100 = 1 + .57 = 1.57

So, for the improper (or mixed) fraction with denominator 100, place decimal point before second digit from the right of the number.

∴ To write fraction whose denominator is multiple of 10 as decimal, we simply count the number of zeros in the denominator and place the decimal point before the same number of digits from the right in the numerator.

Fraction with denominator other than multiple of 10 (other than 10, 100, …)

First we find the equivalent fraction whose denominator is multiple of 10 (i.e., a decimal fraction) and then proceed as above.

11/5 = 11×2/5×2 = 22/10 = 2.2

1/2 = 1×5/2×5 = 5/10 = 0.5

## Comparing Decimals

↬ Decimal with greater integral part will be greater among other decimals.

1.2 > 0.2,

7.2 > 6.2,

42.75 > 8.32

↬ For decimals having equal integral parts, we compare their tenths,

1.24 > 1.12,

7.7 > 7.6,

32.75 > 32.45,

0.2 > 0.0

↬ For decimals having equal integral parts and equal tenths, we compare their hundredths,

1.24 > 1.23,

7.76 > 7.71,

32.75 > 32.76,

0.21 > 0.20

## Addition & Subtraction of Decimals

↪ Write down the decimal numbers, one under the other, with the decimal points lined up.

↪ Add zeros to the beginning or end as required in decimals (not in between) so the numbers have the same length.

↪ Add zeros to integers after putting decimal point to match up the length.

↪ Put the decimal point below in line in the sum .

 00.007 + 08.500 + 30.080 11.600− 09.847 0 are added to match up the length. = 38.587 = 01.753

Ex: Simplify the following.

(i) 200 + 30 + 5 + 2/10 + 9/100

(ii) 16 + 3/10 − 5/1000 = 16 + 0.3 − 0.005 = 16.305

Solution:

(i) 200 + 30 + 5 + 2/10 + 9/100

= 235 + 0.2 + 0.09

= 235 + 0.20 + 0.09

= 235 + 0.29

= 235.29

(ii) 16 + 3/10 − 5/1000

= 16 + 0.3 − 0.005

= 16 + 0.300 − 0.005

= 16 + 0.295

= 16.295

## Some applications of Decimals

↬ Money

100 paise = Re 1

⇒ 1 paise = Re 1/100 = Re 0.01

⇒ 65 paise = Re 65/100 = Re 0.65

⇒ 105 paise = 100paise + 5 paise = Re 1 + Re 0.05 = Re 1.05

↬ Length

100 cm = 1 m

⇒ 1 m = 1/100 m = 0.01 m

⇒ 56 cm = 56/100 cm = 0.56 cm

⇒ 209 cm = 200cm + 9 cm = 2 m + 0.09 m  = 2.09 m.

↬ Weight

1000 g = 1 kg

⇒ 1 g = 1/1000 kg = 0.001 kg

⇒ 56 g = 56/1000 kg = 0.056 kg

⇒ 209 g = 209/1000 kg = 0.209 kg

⇒ 5049 g = 5000 g+ 49 g
= 5 kg + 0.049 kg = 5.049 kg