0
For simplicity I will assume you're working in base x, for any integer x greater than 1, although the argument extends to integers greater than 1 in absolute value (note that in base -1,1 all decimal numbers are in fact integers and that in base 0 decimals are not very well defined). In base x, x can of course be conveniently denoted as 10, so in the remainder of this answer I will work in base x. It is sufficient to show that there exists a decimal number that is not an integer so take 0.1 or 10^-1. This number has the property that 10*0.1 = 1, it is the multiplicative inverse of 10. I will now prove by induction that no positive integer has this property.
Base case: 1*10 = 10 which is greater than 1 by assumption.
Suppose n*10 is greater than 1, then (n+1)*10 = n*10+1*10 = n*10 + 10 which is still greater than 1.
So we now know that n*10 is always greater than 1 for any n greater than 0, from which it can be deduced that for these n, n*10 is also unequal to -1. Therefore, for no integer n unequal to zero can n*10 be 1. Now assume n=0, then n*10 = 0*10 = 0 which is not equal to 1 either.
Thus, no integer n has the property n*10=1, whereas the decimal number 0.1 does. So 0.1 is not an integer and therefore the decimal numbers are not integers.
Still curious? Ask our experts.
Chat with our AI personalities
Rafa
Vivi
ProfessorAdd your answer:
Earn +20 pts
Is decimal numbers considered integers?
No, because integers are whole numbers
Do integers numbers have decimals?
No, integers are whole numbers that do not have any decimal or fractional parts. Integers include positive numbers, negative numbers, and zero. Decimals are used to represent numbers that fall between integers on the number line.
What are examples of non integer numbers?
Non-integers are fractions, decimal numbers, and irrational numbers. Integers are positive and negative whole numbers.
What are natural and integer numbers?
Integers are those numbers that can be written without a fraction or decimal component, such as 1,200,37,-4 Natural numbers are the non negative integers.
Is .75 an integer?
No, integers are whole numbers. Whole numbers are numbers that do not have a fraction or a decimal following them. 7 and 8 would be integers but anything in between is not.
Is decimal numbers considered integers?
No, because integers are whole numbers
Are decimal numbers integers or rational numbers?
They can be integers, rational numbers or even approximations for irrational numbers.
Do integers numbers have decimals?
No, integers are whole numbers that do not have any decimal or fractional parts. Integers include positive numbers, negative numbers, and zero. Decimals are used to represent numbers that fall between integers on the number line.
What are examples of non integer numbers?
Non-integers are fractions, decimal numbers, and irrational numbers. Integers are positive and negative whole numbers.
Are fraction decimal and integers are rational numbers?
yes
Why aren't rational numbers called integers?
because not all rational numbers are integers, recurring numbers, numbers to 1 decimal place and fractions are rational as well but all integers are rational
Is 1.125 a integer?
no, integers are whole numbers and have no decimal points ; 1,2,3 etc. are integers but not 1.8, for example do not have any values after the decimal
When 510510 is converted to a decimal it is a decimal because .?
It is a decimal because it is an integer and all [normal] integers are decimal numbers.
Are integers decimal numbers?
Yes. The decimal part is 0 (for example 16 is 16.0)
Has no fractional parts or decimal parts?
Integers or whole numbers
What are positive and negative numbers but have no decimal or fractional portion?
Integers.
Where are the integers on a number line?
If you have drawn a number line counting in whole numbers, the integers are those whole number points. Any decimal numbers in between are not integers.
