Integers
Whole numbers including negative whole number
Example: …,-4, -3, -2, -1, 0, 1, 2, 3, 4,…
A positive number is any number greater than zero.
Example: 1, 2, ½, 6.4
Negative Numbers
We read -1 as negative one.
A negative number is any number less than zero.
Example: -1, -2.5, -4/7
More Numbers Defintions
Natural numbers are whole numbers equal of greater than 0.
Example: 1 ,2 ,3, 4,…
Rational numbers: a/b where a and b are integers and b is not 0, and terminating
Example: 2/3, 6/7, -4/5, 1.2, -10.8, 200, -430
Irrational numbers: Real numbers that cannot be written as a/b
Example: π, √2
Real numbers: The limit of convergent sequence of rational numbers; rational and irrational numbers.
Prime Number
Greater than 1 that cannot be formed by multiplying 2 smaller natural number.
Example: 2, 3, 5, 7, …
2 = 1 x 2, 13 = 1 x 13
* 1 is not a prime number because prime numbers are greater than 1
* All even numbers are not prime number except 2.
Example
List all the prime number that are greater than 10 and less than 20
Step 1: List all the numbers
11, 12 , 13 , 14, 15, 16, 17, 18, 19
Step 2: Strike out the non-prime number
11, 12 , 13 , 14, 15, 16, 17, 18, 19
Step 3: List out the prime number
11, 13, 17, 19
Lowest Common Factor
The smallest common multiples of two or more numbers
Example
Find the LCM of 8 and 20
2 | 8 20
2 | 4 10
2 5
LCM is 2 x 2 x 2 x 5 = 40
Highest Common Factor (for NA/O)
Largest factor that divides 2 numbers without remainder.
Example
Find the HCF of 12 and 32
2 | 12 32 (smallest common divisible no=2 for 12,32, place result below)
2 | 6 16 ( " " " " = 2 for 6, 16, place result below)
3 8 (no more common divisible number)
HCF = 2 x 2 = 4
Example
Find the HCF of 24 and 30 (Multiplying all factors that appear in both list)
24 = 2 x 2 x 2 x 2 x 3, 30 = 2 x 3 x 5
HCF is 2 x 3 = 6
The smallest common multiples of two or more numbers
Example
Find the LCM of 8 and 20
2 | 8 20
2 | 4 10
2 5
LCM is 2 x 2 x 2 x 5 = 40
Prime factorisation (for NA/O)
Numbers in the form of product of prime numbers
Example: 6 = 2 x 3,
18 = 2 x 3 x 3
Index/Power
The small, raised number next to a normal letter or number, to be multiplied by itself
Example
b2= b x b
43 = 4 x 4 x 4
Square power 2 : 2
- multiplying by itself
Example
(1) Square of 4 = 42 = 4 x 4 = 16
(2) 32 = 3 x 3 = 9
(3) Square of (-5)2 = (-5) x (-5) = 25
Perfect Square are the squares of whole numbers:
(2 x 2) 4, (3 x 3) 9, (4 x 4) 16, …
Square Root - the Opposite of Square
A number that when multiplied by itself, gives the number
Symbol: √
5 --> square 52 --> 25
5 <-- Square root √25 <-- 25
Example
(1) 16 = 4 x 4
√16 = 4
(2) a2 = 25
a = + √5 x 5 = = +5
Why +5 when a number is square-root?
(-5) x (-5) = 25 ( -ve x -ve = + ve)
5 x 5 = 25
=> 25 = (-5)2 = 52 = 25
a2 = + √a x a = -a or a
Cube - Power of 3 : 3
- Multiplying the number by 3 times
Example:
(1) Cube of 3 = 33 = 3 x 3 x 3 = 27
(2) 43 = 4 x 4 x 4 = 64
(3) cube of (-5) 3 = (-5) x (-5) x (-5) = = -125
Cube Root
A value that when ‘cubed’ gives the original number
Symbol: 3√
4 --> cube 43 --> 64
4 <-- cube root 3√64 <-- 64
Example:
3√8 = 3√2 x 2 x 2 = 2
3√216 = 3√6 x 6 x 6 = 6
Practice
1. Write all the factors of the following numbers.
(a) 120 (b) 324 (c) 112
2. Find the prime factorization of the following numbers
(a) 180 (b) 1560 (c) 214
3. Write the prime factorization of 180 x 214
4. Find the HCF of the following pairs of numbers
(a) 114 and 270 (b) 45 and 135
5. Find the LCM of the following numbers.
(a) 55 and 88 (b) 420 and 245
6. The LCM of x and 22 x 33 x 7 is 22 x 34 x 5 x 72 x 11. Find the smallest possible
value of x.
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