Integers 7 Flashcards
Product of 11 negative integers and3 positive integers
Negative
Product of even no of negative no
Positive
X/0 is
Undefined
0/x is
0
Vulgar fractions
Not decimal fractions
Mean is denoted by
X bar
Squares of numbers won’t end in
2,3,7,8
To find the sum of first n natural numbers
n (n+1) / 2
To find the sum of first n square numbers
n (n+1) (2n+1) / 6
In a logarithmic expression the number other than the base and the exponent is called
Argument
Imaginary numbers are represented by
i = √-1
√-2 = √2 i
Who introduced the concept of logarithm for the first time
John Napier
Who prepared a and published the logarithm tables
Henry Briggs
The log of a number has 2 parts
Characteristic and Mantissa
Mantissa - part of the no after decimal place
Characteristic - “ “ “ “ “ before decimal place
We can calculate Log of numbers with a maximum of ………… digits (using log table)
4
The value of mantissa in a log of the number can be calculated by using the
Logarithm table
How can you find antilog
Take the mantissa of the log , and using the table find it’s antilog which is similar to the way in which we find log. After that if the characteristic is 4, then multiply the mantissa with 10⁴
If the log of a number is negative -
E.g. -3.5477 ,it is actually written as
_
3.5477
To find sum of first n perfect cubes
[ n(n+1)/2 ] ²
Common logarithms are represented by
Log N
Natural logarithms are represented by
In n
Common logarithms have base -
10
Natural logarithms have base
e = 2.71
Total no of factors of a number
E.g. Take a number 648
648 = 2³ x 3⁴ Total no of factors for 648 = (3+1) (4+1)
= 4x5 = 20 factors
Total no of prime factors of a number
E.g. Take a number 648
648 = 2³ x 3⁴ Total no of prime factors for 648 = 3 + 4
= 7 factors
Perfect squares contain ………. no of factors
Odd
Relation between HCF and LCM of 2 numbers
Product of 2 numbers = HCF x LCM
HCF and LCM of fractions
LCM =
LCM of numerator / HCF of denominator
HCF =
HCF of numerator / LCM of denominator
When we divide a negative number by a positive number , the remainder cannot not be
Negative
E.g. -22/7 → quotient : -3. Remainder : -1
This is not correct
Instead it should be Q: -4 , R: +6
When a square is divided by 3 or 4 , the remainder is
1 or 0
Geometric progression formula
[a(rⁿ - 1)] / r-1
a - 1st term
r - common ratio
n - total no of terms
Product of n consecutive numbers is
n!
Total no of terms in (a + b)ⁿ
n + 1
Total sum of coefficients in (a + b)ⁿ
n²
0 ! = ….! = ….
0! = 1! = 1
Sum of odd binomial coefficients =
= Sum of even binomial coefficients = 2 ⁿ⁻¹
Every prime number greater than 3 can be expressed as
6k +/- 1
(xⁿ - aⁿ) is divisible by …….. for all values of n
(x - a)
(xⁿ - aⁿ) is divisible by …….. for all even values of n
(x - a) and (x+a)
(xⁿ + aⁿ) is divisible by …….. for all odd values of n
x + a
Divisibility rule of 7
Number - (2 x last digit) = a no divisible by 7
Divisibility rule of 13
Number + (4 x last digit) = a no divisible by 13
Divisibility rule of 17
Number - (5 x last digit) = a no divisible by 17
Divisibility rule of 19
Number + (2 x last digit) = a no divisible by 19
Fermat’s little theorem
states that if n is a prime number, then for any integer a, the number aⁿ− a is an integer multiple of n.
Father of numbers
Pythagoras
Father of Indian mathematics
Aryabhatta
Integers are represented by
Z or I
Twin primes
Pair of prime numbers with only composite no between them.
E.g. 3,5 ; 5,7
Complex numbers
Numbers with 2 terms in which one term is an imaginary number.
Irrational numbers represented by
Q’ or T
Decimal fractions
Fractions having denominator as 10
Common decimals
Fractions having denominator not as 10
Compound fractions
Fractions whose both numerators and denominators are fractions.
Mixed recurring decimal
A recurring decimal in which at least one digit doesn’t repeat.
Bar bracket also called
Vinculum
Total no of terms in (a + b)ⁿ
n + 1
Total sum of coefficients in (a + b)ⁿ
n²
0 ! = ….! = ….
0! = 1! = 1
Sum of odd binomial coefficients =
= Sum of even binomial coefficients = 2 ⁿ⁻¹
Every prime number greater than 3 can be expressed as
6k +/- 1
(xⁿ - aⁿ) is divisible by …….. for all values of n
(x - a)
(xⁿ - aⁿ) is divisible by …….. for all even values of n
(x - a) and (x+a)
(xⁿ + aⁿ) is divisible by …….. for all odd values of n
x + a
Divisibility rule of 7
Number - (2 x last digit) = a no divisible by 7
Divisibility rule of 13
Number + (4 x last digit) = a no divisible by 13
Divisibility rule of 17
Number - (5 x last digit) = a no divisible by 17
Divisibility rule of 19
Number + (2 x last digit) = a no divisible by 19
Fermat’s little theorem
states that if n is a prime number, then for any integer a, the number aⁿ− a is an integer multiple of n.
Father of numbers
Pythagoras
Father of Indian mathematics
Aryabhatta
Integers are represented by
Z or I
Twin primes
Pair of prime numbers with only composite no between them.
E.g. 3,5 ; 5,7
Complex numbers
Numbers with 2 terms in which one term is an imaginary number.
Irrational numbers represented by
Q’ or T
Decimal fractions
Fractions having denominator as 10
Common decimals
Fractions having denominator not as 10
Compound fractions
Fractions whose both numerators and denominators are fractions.
Mixed recurring decimal
A recurring decimal in which at least one digit doesn’t repeat.
Bar bracket also called
Vinculum
Founder of modern number theory
Pierre de Fermat
‘Mathematics is the queen of the sciences and number theory the queens of mathematics’ was said by
Carl Friedrich Gauss
Most ancient device used for calculation purposes
Abacus
Highest power of x (prime number) dividing a factorial y
= [y/x] + [y/x²] + [y/x³] ……
[ ] represents that the integral value is only taken
To convert octal to binary
Convert each digit in the octal number into 3 binary digits. Replace it , and u have converted it
While converting decimal system into another system the result is taken from the …… direction
Down to top
sum of the first n cubes, 1³ +2³ +…+n³ =
(n(n+1)/2)²
