The number system or the numeral system is the system of naming or representing numbers. There are various types of number systems in maths like binary, decimal, etc. This lesson covers the entire concepts of the numeral system with their types, conversions and questions.
Numbers are of many types. These can be defined as given below: Fig: Classification of Numbers.
SOME PROPERTIES OF NUMBERS:
The numbers which can be found on the number line and include both rational and irrational numbers are known as real numbers, e.g., -1.5,√2,0,1,2,3,π.Almost any number which you can imagine is a real number.
COMPARISON OF TWO RATIONAL NUMBERS:
Ascending and descending orders of rational numbers.
Steps to write two or more rational numbers in ascending or descending order.
Step 1: Write the given rational numbers with +ve denominator.
If given rational number are :
Step 2 :
Find the LCM of denominators.
LCM of 10, 8, 3 and 5 is 120.
Step 3: Write all the given rational numbers with this LCM as denominator so that the numbers so written are equivalent to the given rational numbers.
Step 4: Now compare the , so found equivalent rational numbers on the basis of their numerators.
Since -96 < -84 <- 80 < -75
Second method to decide which rational number is greater and which one is smaller.
Let the rational numbers to be compared are
Step I : Cross multiply the numbers i.e.
We get 21 and 25.
Out of 21 and 25, 21 is smaller.
Step II : Find out that this 21 is formed with the help of which numerator. It is 3 of 3/5
Step III : So 3/5 is the smaller rational no.
Note: If the numbers to be compared are three or more, compare the rational numbers by taking pairs.
MULTIPLICATIVE INVERSE OF A RATIONAL NUMBER:
The multiplicative inverse of a non-zero rational number
Also for every non-zero rational number a/b , there exists a rational number
If the product of two rational numbers is 1, then we say that one number is the multiplicative inverse of the other.
Caution: There is no multiplicative inverse for the rational number zero.
Prime numbers: If there is a number which has no divisor apart from 1 and itself, it is called a prime number.
Some characteristics of prime numbers.
1) There are 25 prime numbers between 1 and 100 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.
2) 2 is the only even prime number.
3) Between 90 and 100, 97 is the only prime number.
4) 1 is not a prime number. Nor it is a composite no.
To test whether a given number is prime or not.
Step 1. Let the number to be tested be x.
Step 2. Take an integer slightly more than the sq. root of x say y.
Step 3. Test if the given number (x) is divisible by any of the prime numbers less than y
Step 4. If it is not divisible by any of them, then the number x is a prime number., otherwise it is a composite number
Example: Is 377 a prime number?
Solution: 19 is the approximate square root of 377. Prime numbers less than 19 are 2, 3, 5, 7, 11, 13, 17, 377 is divisible by 13.
So, 377 is a composite number.
Difference between face value and place value:
Face value of a digit in a numeral:
The face value of a digit remains as it is and does not change, what ever place it may occupy in the place-value chart. Thus, the face value of 7 is always 7 where ever it may be.
Place value of a digit in a numeral:
The place value of a digit in a numeral depends upon the place it occupies in the place-value chart.
If 5 occurs at one's place, its place value = 5 ones = 5 x 1=5
If 5 occurs at ten's place its place value = 5 x 10=50
If 5 occurs at hundreds place, its place value= 5 hundreds = 5 x 100=500 and so on.
Example: Consider the numeral 5439602
Solution: In this numeral
Place value of 2 is = 2 ones = 2 x 1=2
Place value of 0 is = 0 tens = 0 x 10 = 0
Place value of 6 is = 6 hundreds = 6 x 100 = 600
Place value of 9 is = 9 thousands = 9 x 1000 = 9000
Place value of 3 is = 3 ten thousands = 3 x 10000 = 30,000
Place value of 4 is = 4 lakhs = 4 x 100000 = 400000
Place value of 5 is = 5 ten lakhs = 5 x 10,00000 = 50,00000
unit digit = 3167
31→ 3 → 3
32 → 9 → 9
33 → 27 → 7
34 → 81 → 1
This cycle will continue
⇒ divide the power of 3 by 4
167/4 ⇒ remainder is 3
33 ⇒ 7
unit digit = 1 × 7 = 7