A B C D E F G H I J K L M N O P Q R S T U V W X Y Z The counting numbers: 1, 2, 3, 4, and so on.
Precisely: A member of ℕ = {1, 2, 3, …}, the numbers used for counting (some books include 0).
How it works Each natural number is one more than the one before. People counted long before writing, by matching objects with tally marks, as on the Ishango bone.
Examples The number of students in a class is a natural number . 7 is a natural number , but 0, −3 and 1/2 are not (in this book). The sum of the first n natural numbers is n(n + 1)/2. Do not confuse: Books differ on whether 0 is a natural number; this book starts at 1. Whole numbers are 0, 1, 2, …
Used in counting, sequences, the building block of all other number sets Symbols ℕ Related integer , one-to-one correspondence , zero From Class 9 Mathematics, Chapter 3: The World of Numbers Sources Our chapter: class-9/mathematics/03 OpenStax: Intermediate Algebra 2e, 1.1 Key Terms (counting numbers) A formula that gives any term of a sequence from its position number n.
Precisely: The term tₙ in position n, usually given by an expression in n, such as tₙ = a + (n − 1)d for an AP or tₙ = arⁿ⁻¹ for a GP.
How it works Put n = 1, 2, 3, ... into the formula to get the terms. To test whether a number is a term, set the formula equal to it and solve for n: it is a term only when n comes out a natural number.
Examples The nth term of the odd numbers is 2n − 1, so the 53rd is 105 and 137 is the 69th. With nth term 5n − 2, 308 is the 62nd term, but 471 is not a term (n would be 94.6). With nth term 3n − 7, 557 is the 188th term. The nth term of 3, 6, 12, 24, ... is 3 × 2ⁿ⁻¹. Do not confuse: n is the position, the term is the value: in t₅ = 25, n = 5 and the term is 25. A position must be a natural number (1, 2, 3, ...).
Used in APs and GPs, checking whether a number belongs to a sequence, counting problems Related explicit formula (explicit rule) , sequence , term , subscript (tₙ notation) From Class 9 Mathematics, Chapter 8: Predicting What Comes Next: Exploring Sequences and Progressions Sources Our chapter: class-9/mathematics/08 OpenStax: College Algebra 2e, 9.1 Sequences and Their Notations (general term) number line NUM-ber lyn Class 9
A straight line with 0 marked on it, where every number has its own point: positives to the right, negatives to the left.
Precisely: A line on which each real number corresponds to exactly one point, measured by its distance and direction from 0.
How it works Choose a point for 0 and a length for 1. Then 2 is two lengths right, −3 three lengths left, and 1/4 a quarter of a length right.
Examples On the number line , −4 lies to the left of −1, so −4 is the smaller number. To place 9/4 on the number line , split the gap from 2 to 3 into four equal parts and take one. The rational numbers cover the number line closely but leave gaps at points like √2. Do not confuse: The number line is one-dimensional; the coordinate plane uses two number lines at right angles.
Used in comparing numbers, absolute value and distance, the axes of the coordinate plane Related integer , real number , absolute value , axis From Class 9 Mathematics, Chapter 3: The World of Numbers Sources Our chapter: class-9/mathematics/03 OpenStax: Prealgebra 2e, 1.1 Key Terms (number line) The top number of a fraction: it counts how many of the equal parts are taken.
Precisely: In the fraction p/q, the number p, the dividend.
How it works With the denominator fixed, a bigger numerator means a bigger fraction. Fractions with the same denominator are added by adding their numerators.
Examples In 3/8 the numerator is 3: three of the eight parts. 2/7 + 3/7 = 5/7: add the numerators and keep the denominator. A fraction with numerator bigger than its denominator, like 7/4, is more than 1 (an improper fraction, 1 3/4). Do not confuse: Add numerators only when the denominators are the same: 1/2 + 1/3 is not 2/5.
Used in fractions, probability (the number of favourable outcomes) Related denominator , fraction From Class 9 Mathematics, Chapter 3: The World of Numbers Sources Our chapter: class-9/mathematics/03 OpenStax: Prealgebra 2e, 4.1 Visualize Fractions (numerator)