Numbers And Operations Flashcards
Natural number
Greater than 0 and has no decimals or fractions attached.
Whole number
Natural numbers and the number 0
Integers
Positive and negative natural numbers and 0
Rational number
Can be represented as a fraction.
Irrational number
Cannot be represented as a fraction. Never ends or resolves into a repeating pattern.
Real number
Can be represented by a point on a number line.
Imaginary number
Imaginary numbers produce a negative value when squared.
Complex number
All imaginary numbers are complex. Real numbers. A + Bi
Factor
All numbers that can multiply together to make the number
Composite number
More than two factors. Example: 6. Factors 1, 6, 3, 2
Commutative property
An operation if order doesn’t matter when performing the operation
For example: (-2)(3) = (3)(-2)
Associative property
An operation if elements can be regrouped without changing the result.
For example: -3 + (-5 + 4) = (-3 + -5) + 4
Distributive property
A product of sums can be written as a sum of products.
For example: a(b+c) = ab + ac
FOIL
First, Outer, Inner, Last
Useful way to remember the distributive property
Identity element
The identity element for multiplication on real numbers is 1 (a x 1) = a
For addition is 0 (a + 0) = a
Inverse element
Addition : -a because a + (-a)=0
Multiplication: 1/à because a*1/a=a
Closed number system
An operation on two elements of the system results in another element of that system.
For example: integers during addition, multiplication, subtracting, but not division. Dividing two integers could result in a rational number that is not an integer.
Conjugate
where you change the sign (+ to −, or − to +) in the middle of two terms.
Examples:
• from 3x + 1 to 3x − 1
• from 2z − 7 to 2z + 7
• from a − b to a + b
Complex conjugate
the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.
4+7i is 4 - 7i.
Adding complex numbers
simply add the real parts and add the imaginary parts. For example:
(3+4i)+(6−10i)
= (3+6)+(4−10)i
= 9−6i
Subtracting complex numbers
we simply subtract the real parts and subtract the imaginary parts. For example:
=(3+4i)−(6−10i)
=(3−6)+(4−(−10))i
=−3+14i
Multiplying complex numbers
we perform a multiplication similar to how we expand the parentheses in binomial products:
(a+b)(c+d)=ac+ad+bc+bd
Unlike regular binomial multiplication, with complex numbers we also consider the fact that
i^2=−1
2⋅(−3+4i)
2⋅(−3+4i)
=2⋅(−3)+2⋅4i
=−6+8i
3i⋅(1−5i)
3i⋅(1−5i)
=3i⋅1+3i⋅(−5)i
=3i−15i ^2
=3i−15(−1)
=15+3i
(2+3i)⋅(1−5i)
=2⋅1+2⋅(−5)i+3i⋅1+3i⋅(−5)i
=2−10i+3i−15i ^2
=2−7i−15(−1)
=17−7i
Scientific notation
Learn how to add and subtract in scientific notation
Absolute value
The distance the number is from zero
|-2| = 2
metric system mnemonic device: King Henry Drinks Under Dark Chocolate Moon
Kilo
Hecto
Deca
Unit
Deci
Centi
Milli
Factorial
N is denoted by n! and is equal to 1x2x3x4x….xn.
0! And 1! = 1
Find the whole in a ratio
Add the values in the ratio
Ex: 2:3, whole is 5
Proportion
An equation that states two ratios are equal
A/B = C/D where a and d terms are the extremes, and the b and c are the means
Radicals
Expressed as b square root of a. B is callled the index and a is the radicand
Used to indicate inverse operation of an exponent: finding the base which can be raised to b to yield a.
For example, 3 square root 125 is equal to 5 because 5x5x5=125
Matrix
Rectangular arrangement of numbers into rows (horizontal set of numbers) and columns (vertical set of numbers)
Rows (matrix)
Horizontal set of numbers
Columns (matrix)
Vertical set of numbers
Square matrix
Same number of rows and columns
Dimensions of a matrix
M x N
M= number of rows
N= number of columns
Determinants of a matrix
Written as det(A) or |A| is a value calculated by manipulating elements of a square matrix.
Identity matrices (I)
A square matrix with values of 1 forming a diagonal from the upper left corner to the bottom right corner; the rest of the elements are 0
Inverse matrix (A ^-1)
A square matrix that, when multiplied by the original matrix, results in the identity matrix (A x A^-1 = 1)
Arithmetic series
The sum of an artithmetic sequence
Geometric series
The sum of geometric sequence
Arithmetic growth
Constant growth, meaning that the difference between any one term in the series and the next consecutive term will be the same constant. This constant is called the common difference.
To list the terms in the sequence, one can just add or subtract the same number repeatedly.
Recursive definition
Next term = current term + common difference
Geometric sequence
Similar to an arithmetic sequence, but with multiplication instead of addition
Converge (infinite geometric sequence)
If the infinite terms of the sequence add up to a finite number
Diverge (infinite geometric sequence)
If the infinite terms of the sequence add up to an infinite number
