Maths Chapter 1 - Real Numbers

rational numbers

Q = {any number that can be written in the form a/b, where a,b ∈ Z, b ≠ 0}. Rational numbers are also called fractions.

integers

Z = {...-4,-3,-2,-1,0,1,2,3,4...}

factor

a factor of a natural number is any number that divides evenly into the given number. 1 is a factor of every natural number. every natural number is a factor of itself.

multiple

a multiple of a natural number is itself a natural number, into which the natural number divides, leaving no remainder.

scientific notation

a number is written in scientific notation if it is of the form a × 10*n, where 1 < a ≤ 10 and n ∈ Z. another name for scientific notation is standard form.

order of magnitude

a number rounded to the nearest power of 10 is called an order of magnitude. 1 = 10*0. When finding the difference between two numbers written in scientific notation, if the decimal number is less than 5, round to 1, if not, round to 10.

irrational numbers

an irrational number is a number that can't be written in the form a/b, where a,b ∈ Z, b ≠ 0. Irrational numbers can't be written as fractions.

properties of integers

closure property: a + b and a × b = Z commutative properties: a + b = b + a and a × b = b × a associative properties: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). distributive property: a × (b + c) = (a × b)+(a × c) identity elements: a + 0 = a and a × 1 = a. additive inverse: for every integer a there exists an integer -a, such that a + (-a) = 0. -a is the additive inverse of a.

composite number

composite numbers are natural numbers greater than 1 which are not prime numbers. the first five composite numbers are 4,6,8,9 and 10.

the fundamental theorem of arithmetic

every natural number greater than 1 is either prime or can be written as a unique product of primes.

factorial

let n be a natural number. n! read as n factorial, is the product of the natural numbers 1,2,3...n. We define 0! to be 1.

prime number

prime numbers are natural numbers with only two factors, themselves and 1. 2 is the only even prime number. 1 is not a prime as it only has 1 factor, itself. 0 is not a prime number as it has an infinite number of factors. Euclid proved the number of primes was infinite.

proof by contradiction

proof by contradiction is a form of proof that establishes the truth of a proposition by showing that the proposition being false would imply a contradiction.

highest common factor

the highest common factor of two natural numbers, n^1 and n^2, is the largest natural number that divides evenly into both n^1 and n^2.

lowest common multiple

the lowest common multiple of two numbers is the smallest multiple that both numbers share.

natural numbers

the natural numbers are the ordinary counting numbers. N = {1,2,3,4...}.

properties of rational numbers

the properties of integers hold true for rational numbers, all integers are rational numbers. two fractions are equivalent if they have the same value, e.g.1/2 =3/6. the reciprocal of a fraction is found by inverting the fraction, e.g. 11/12 is the reciprocal of 12/11. every fraction, except zero, has a multiplicative inverse (reciprocal), and their product is always 1, e.g. 3/4 × 4/3 = 1.

real numbers

the real numbers can be thought of as the set of all numbers that lie along an infinitely long number line. the rational numbers along with the irrational numbers make up the real number system.

significant figures

the significant figures of a number express a magnitude to a specified degree of accuracy. leading zeros are not significant figures. zeros that appear between two non-zero digits are significant.

examples of significant figures in regard to 0.0053 and 503,25

0.0053 to 1 significant figure = 0.005 503.25 to 4 significant figures = 503.3