Numbers
Vocabulary: natural number - whole number - integer - decimal - rational - real number number line - standard form - scientific notation - decimal place - significant figure to correct to - to round to - digit - prime number - factor - multiple - prime factor form common factor - Greatest Common Factor (GCF) - Highest Common Factor (HCF) co-prime numbers - Lowest Common Multiple (LCM) • Sets of numbers o
Natural numbers (or whole numbers, or integers) are the set of numbers 0 ; 1 ; 2 ; 3...as used in counting.
It is a matter of choice whether 0 is included or not. In France 0 is included in the set of natural numbers. In Great Britain 0 is not included in the set of natural numbers. ℕ is the symbol to indicate that the set of natural numbers is to be used. o
Integers are numbers made from the natural numbers (including 0) by putting a positive or a negative sign in front : ..; -2 ; -1 ; 0 ; 1 ; 2 ; 3 , ...
ℤ denotes the set of integers o o o
Decimals can be written in the form --- where a is an integer and n is a natural number. Ⅾ denotes the set of decimals. Rational can be written in the form --- where a is an integer and b is a natural number not equal to zero. ℚ denotes the set of rational numbers. The set of real numbers is made of all the numbers we know in year 11. ℝ denotes the set of real numbers.
Examples: -4.5 ; 1 ; 0.0909090909... ; 8 are all rational since they can be re-written as: -9/2 ; 1/1 ; 1/11 ; 8/1
• Number line: A number line is a graduated straight line along which it is possible to mark all the real numbers. 0
1
--------------------------------------!---------!----------------------------------• Standard form When numbers become very large or very small, standard form (or scientific notation) is used to make the numbers more manageable. To write a positive number in standard form , you express it as the product of a number between 1 and 10 (not including 10) and a power of 10. Decimal places and significant figures o
Decimal places (d.p.) refer to the digits to the right of the decimal point, and significant figures (sig.fig. or s.f.) include all the digits in a number to the right of the first non-zero digit (0.0026 has two significant figures: 2 and 6) or at the end of a number to the left of the decimal point (3600 has two significant figures : 3 and 6). In working with numbers, too many figures are often in an answer. These are reduced by giving answers corrected or rounded to an appropriate number of decimal places or significant figures.
o
Rounding or correcting consists of discarding any digits beyond the specified number and adjusting the last digit retained where necessary.
Rules for rounding: o o
If the first digit to be discarded is les than 5, leave the last digit unaltered. If the first digit to be discarded is greater than or equal to 5, round up by 1 the last digit retained.
• Prime numbers o When you write 56 = 8 7 or 56 = 4 14 , you have factored 56. In the first case the factors are 8 and 7. In the second case the factors are 4 and 14. o 56 is a multiple of 8 o A prime number, or prime, is an integer greater than 1 which has only two factors. The first ten prime numbers are : 2, 3, 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29. o To find the prime factor form of a positive integer, you express it as a product of primes. Exponents are generally used for prime factors that occur more than once in a factorization. The prime factor form of an integer is unique (there is only one) except for the order of the factors.
o A factor of two or more integers is called a common factor of the integers. o The greatest common factor (GCF) (or highest common factor (HCF) of two or more integers is the greatest factor of all the given integers. o Two numbers which have a greatest common factor of 1 are called co-prime numbers. o The lowest common multiple (LCM) of several numbers is the lowest number that is a multiple of all the given numbers. (This is useful when you are adding or comparing fractions).
Exercise 1: Evaluate, giving your answers in standard form: a) 2.8
10 + 1.6
10 = ..................................................................................................
b) 2.5
10 -1.6
10 = ....................................................................................................
Exercise 2: Fill in the blanks. 2.7243 is ............... to 2 decimal places. 2.7283 is ............... to 2 d.p. 2,7253 is ............... to 2 d.p. 2.7243 is ................ to 2 significant figures. 2.78 rounds to ................ to 2 s.f. 827 rounds to ................. to 2 s.f. 999 rounds to ................. to 1 s.f.
Exercise 3: State whether or not the number is prime. Give the prime factor form of the number. 31 - 32 46 - 51 - 81
Exercise 4: Find the GCF of each pair of numbers, by using the prime factor form of both integers. 15 and 25 - 22 and 35 - 23 and 46
