Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(35-42,\) find the multiplicative inverse of each number. $$4$$

Step-by-Step Solution

Answer

The multiplicative inverse of 4 is \(1/4\) or \(0.25\).

Step 1: Understanding the concept

A multiplicative inverse or reciprocal for a number x, denoted by \(1/x\) or \(x^{-1}\), is a number which when multiplied by x yields the multiplicative identity, 1.

Step 2: Finding the multiplicative inverse of 4

The multiplicative inverse of a number is found by dividing 1 by the number. So, for the number 4, divide 1 by 4. This results in \(1/4\) or \(0.25\). Therefore, \(1/4\) or \(0.25\) is the multiplicative inverse of 4.

Key Concepts

Algebraic Concepts

Algebra might seem complex at first, but it's really about finding patterns and solving puzzles. Just like a detective looking for clues! In algebra, we often work with unknown values, which we call variables. These are often represented by letters like 'x' or 'y'. But unlike a detective mystery, in algebra, rules guide us to the solution.

One of these rules involves the concept of a multiplicative inverse. Picture this: you have a magic number that turns any number you multiply it by, back into 1. This isn't a fairy tale – in algebra, we actually have such magic numbers, and they keep the balance in equations. Imagine if we didn't have these; our math problems would be chaotic, much like a detective story with no resolution. The multiplicative inverse is just one of the cool tools in our algebra toolkit to maintain order and find our 'whodunit' – the value of 'x'.

Reciprocal of a Number

Now let's tackle the 'reciprocal of a number'. It's much simpler than it sounds. Think of a slice of pie. The whole pie is '1', and the slice is a part of that pie. If you have four slices, that's your number '4'. But what if I told you there's a way to use those slices to reconstruct the whole pie? The 'reciprocal' is that magical step. For any slice, or number, it's just a flip! You flip the top and bottom numbers – in math language, you flip the numerator and denominator. If you don't see a bottom number, like with our '4', it's really '4/1'. Flip it, and you get '1/4'. That's the reciprocal.

When you multiply a number by its reciprocal, you get the whole pie – '1'. It's like a math handshake; they meet, agree, and poof – unity! The reciprocal is the ultimate team player in numbers, ensuring that everything balances out perfectly.

Multiplicative Identity

Lastly, let's dive into the 'multiplicative identity'. What's an identity? In life, it's what makes you, you! In math, it's similar – it's what makes a number stay the same, stay itself. When you think about identity in math, think of the number '1'. It's the neutral, the unchanged, the 'I am who I am' of numbers.

Why is '1' so special? Because it's the only number that, when used in multiplication, keeps other numbers exactly as they are. Multiplying by '1' is like looking in the mirror; you see yourself unchanged. This is why we call '1' the multiplicative identity – it's the one constant in the changing world of math.