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Mathematics D 4024

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Unit # 1 – Numbers

Key points

Natural numbers (N)

In mathematics, the natural numbers are those used for counting (as in 'I have seven coins in my pocket') and ordering (as in 'this is the third largest store in the city'). The natural numbers are positive numbers without any decimal or fractional part and start with 1. The set of natural numbers represent mathematically by the set N= {1, 2, 3,4, 5, …}.

Whole number (W)

These are same as natural numbers including the zero. The set of whole numbers represent mathematically as W= {0, 1, 2, 3, 4, 5, …}.

Integers (Z)

These are same as whole number, but negatives are also included. It is represented as Z= {… -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}.

Even numbers (E)

Even numbers are integers which can be completely divided by two, means if an integer is divided by 2 and yields no remainder then it is even number. Even numbers are both positive and negative (including zero, because 0 ÷ 2 = 0 (no reminder)). It is represented as E= {0, ±2, ±4,±6, …}.

Odd numbers (O)

Odd numbers are integers which can’t be evenly divided by two, means if an integer is divided by 2 and yields 1 as remainder then it is odd number. Odd numbers are both positive and negative. It is represented as O= {±1, ±3, ±5, ±7, …}.

Prime and composite numbers

prime number (or a prime)is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product1 × 5 or 5 × 1, involve5 itself. However, 6 is composite because it is the product of two numbers (2 × 3) that are both smaller than 6.


Prime = {2, 3,5, 7, 11, 13, 17, …}

Composite = {4,6, 8, 9, 10, 12, 14, 15, 16, …}

Note: the number 1  is neither a prime nor a composite number because it has only one factor. 

Rational numbers (Q)

In mathematics, a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. A rational number must have a terminating or repeating sequence (after the decimal point), like ¾ is equals to 0.75 so it is rational number, and 1/3 is equals to 0.3333333… hence it is a rational number.



Irrational numbers

All real number which are not rational are irrational numbers. It has nonterminating and non-repeating sequence after the decimal point. 


Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ,and the square root of two; in fact, all square roots of natural numbers, other than of perfect squares, are irrational.


Surds

When we can’t simplify (compute) a square root (or cube root or nth root), then it is a surd. The √2 can’t be simply computed so it is surd, while √4 is 2, so it is not a surd.

Real numbers (R)

The rational and irrational numbers combine called real numbers.


Directed number

Directed number can be positive as well as negative. The sign indicates a direction. E.g. -10m from sea level means 10 meters below from sea level, and +10m rom sea level means 10 meters above the sea level.

Absolute value of a number

The absolute value of modulus |x| of a real number x is the non-negative value of x without regard of its sign. The absolute value of 3 is 3 and absolute value of -3 is 3.

e.g. | 5-2 | = 3 and | 2-5 | = 3

Factors of a number

Factors are numbers we can multiply together to get another number. For example, 4 and 7 are factor of 28, because4 x 7 = 28. A number can have many factors, like number 16 factors are 1, 2, 4,8, 16, all these numbers divide number 16 completely.

Integral Multiples of a number

All the number that we get after multiplying a number with integers are called integral multiples of that number. For example, the integral multiple of 4 are {0, ±4, ±8, ±12, ±16, …}

Prime factors

Any composite number can be represented as product of prime numbers. These numbers are called prime factors of that number. For example, 9 = 3 x 3 (3 and 3 are prime factors of 9), and 80 = 2 x 2x 2 x 2 x 5 (2, 2, 2, 2, 5 are prime factors of 80).

Highest common factor (HCF)

The largest of the common factors of two or more numbers is called the HCF of these numbers. In order to find he HCF of two or more numbers, find the prime factors of all numbers, then select all the common factors, the product of these common factors is HCF.

For example, HCF (540, 294)

540 = 2x2 x3x3x3 x 5 and 294= 2x 3 x 7x7

So, the common factors are 2 and 3, and their product is 6.

So, HCF (540, 294) = 6  

Lowest common multiple (LCM)

The smallest of the common multiples of two or more numbers is called the LCM of these numbers. In order to find he LCM of two and more numbers, find the multiples of both numbers, until we got the same number.

For example, LCM (6, 8)

Multiples o six are 6,12, 18, 24, 30

Multiples of 8 are 8, 16, 24, 32

24 is the first common multiple in both, so it is the LCM.

An interesting fact is The product of two numbers is equals to the product of their LCM and HCF.

p x q = LCM (p, q) x HCF (p, q) 


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