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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, …}.

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

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 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 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 = {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.

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.

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.

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.

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 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.

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, …}

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).

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

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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