Table of Contents

- 1 Do real numbers form a group under multiplication?
- 2 Are real numbers closed under addition and multiplication?
- 3 Is R+ A group under multiplication?
- 4 What types of numbers are closed under multiplication?
- 5 Which of the following is not a group set of natural numbers under multiplication?
- 6 Is Z5 a group under multiplication?
- 7 Is the set of natural numbers under addition a group?
- 8 Is integers under multiplication a group under multiplication?

## Do real numbers form a group under multiplication?

Question: Is the set of real numbers a group under the operation of multiplication? My professor answered it by saying: No. There is no identity element (1*0=0).

## Are real numbers closed under addition and multiplication?

Real numbers are closed under addition, subtraction, and multiplication. That means if a and b are real numbers, then a + b is a unique real number, and a ⋅ b is a unique real number. For example: 3 and 11 are real numbers.

**Is the set of real numbers under the operations of addition and multiplication a field?**

A field is the name given to a pair of numbers and a set of operations which together satisfy several specific laws. A familiar example of a field is the set of rational numbers and the operations addition and multiplication.

**Which of the following is not a group under addition operation?**

Answer: No. The set of positive integers is not a group under the operation of addition.

### Is R+ A group under multiplication?

R+ and Q+ are groups under multiplication, because the product of two positive numbers is positive, and the reciprocal of a positive number is positive.

### What types of numbers are closed under multiplication?

Answer: Integers and Natural numbers are the sets that are closed under multiplication.

**When two real numbers are multiplied the result is still a real number?**

Closure Property The product of any two real numbers will result in a real number. This is known as the closure property of multiplication. In general, the closure property states that the product of any two real numbers is a unique real number.

**Are complex numbers a group under multiplication?**

The set of integers under ordinary multiplication is NOT a group. The subset {1,-1,1,-i } of the complex numbers under complex multiplication is a group.

## Which of the following is not a group set of natural numbers under multiplication?

Part c) The set of natural numbers with multiplication is not a group, since there is no inverse of 2: The identity is 1, so 2*x = x*2 = 1, where x is the inverse. 2x = 1 implies x = 1/2 which is not in the set of natural numbers.

## Is Z5 a group under multiplication?

The set Z5 is a field, under addition and multiplication modulo 5. To see this, we already know that Z5 is a group under addition.

**Is the set of real numbers a group under multiplication?**

So taking this in view, the set of real numbers is not a group under multiplication because the element 0 has no inverse in that group, as division by 0 does not make any sense. However, if you remove 0 from the set of real numbers then the resulting set will be a group with respect to multiplication.

**Are the real numbers a group?**

The real numbers are a group, if addition is used as the operation. (Only the nonzero real numbers with multiplication are a group.) This can be seen by checking all four properties. For addition, the identity is 0 and the inverse of x is − x; for multiplication, the identity is 1 and the inverse of x is 1 x.

### Is the set of natural numbers under addition a group?

4) The set of natural numbersunder additionis nota group, because it does notsatisfy allof the groupPROPERTIES: it does nothave the IDENTITY PROPERTY (see the previous lectures to see why). Therefore, the set of natural numbersunderadditionis not a group!

### Is integers under multiplication a group under multiplication?

10) The set of integers under multiplication is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the INVERSE PROPERTY (see the previous lectures to see why). Therefore, the set of integers under multiplication is not a group! Furthermore, is Za group under multiplication?