Something that may help is changing how you think about 1, 2, 3...
You are not "Tony." Hopefully you're more than that. You're probably a person (though its possible you could be an AI). A more accurate wording would be "Your name is Tony."
Likewise, numbers could use a little freedom. Think of them as "the number called 1" "the number called 2" and so forth. The number is its own thing, we just use symbols like 1 and 2 to represent them. This actually becomes a big deal in programming: The number you call 10 is often called 0x0A by programmers (for reasons well beyond this question). Whether we call it 10 or 0x0A, it's the same number. Add 10+10 and you get the number we call 20. Programmers might call it 0x14, because we're confusing that way, but it's all just names. Nothing more!
We can do some division here also, we can divide 10/5 and get 2 (or more accurately, the number called 2). You can think of 10/5 as a way to construct the number called 2. 10/5 and 2 are representations of the same number -- we just call the number 2 because that's easier on our heads to give numbers just one good name. "One fifth of ten" is just as good of a name for that number, but its longer and harder to remember.
Now what about your fractions, like 1/2. They are also a way to construct a number, but it doesn't have an easy name. We just know the number constructed by 1/2 is "the number half way between zero and one," if we're confident such a number exists in the first place. It's the same number as we construct with 2/4, "the number one quarter of the way between zero and two." Now, as before, we want to make things easy on ourselves. 1/2 happens to be the simplest way to construct that particular number (2/4, 3/6, and many others also construct the same number). We declare the name of that number to be named "1/2".
There are two final questions left to answer. One is whether such a number exists in the first place. That really depends on what you're dividing. If you have a box, and are told "divide it into two equal halves," it might be a bit hard to do so. But if someone opens the box, and inside are a thousand tiny pebbles, and you are told to divide this into two equal halves, you can do it (500 pebbles in each pile). On other other hand, if King Solomon orders you to cut your baby in half to solve a squabble, you would righteously argue that "a baby divided into two equal halves" doesn't quite make sense.
Accordingly we don't always allow fractions when doing math. There are some problems restricted to "integers" (..., -3, -2, -1, 0, 1, 2, 3, ...), where 1/2 actually has no value. However, lots of problems deal with things that can be split into any sized fractions, which we call "rationals". (There are also "real" numbers, which is a very loaded name, but you'll learn about them later in your career).
The final question is "is this useful." Math is just making models. I can summon up some symbols and declare "4 $ Wobble = bleep!" and not be wrong. It's just not a very useful model. I won't be able to communicate much to other mathematicians with it.
Consider if you had a pile of 1000 pebbles. You're told to multiply that number by four. You dig deep into your bag of spare pebbles and pour them out on the table until you have 4000 pebbles. Now you are told to divide it by four, so you take away pebbles until you have 1000 pebbles left.
Now its done in the other order. You start with 1000 pebbles, and you are told to divide that number by four, and remove the rest. You comply, leaving a pile of 250 pebbles. Now you are told to multiply that number by four, so you add more pebbles to do so. You end up with a pile of 1000 pebbles.
In both cases, you end up with the same number of pebbles as you started. In symbols, 1000*4/4 = 1000 and 1000/4*4 = 1000
Now lets do the same with smaller numbers, and to make it interesting, we're going to force it to be an integer problem by using babies again. Forgive me for any graphic imagery which may follow.
You start with one baby. You are told to multiply the number of babies by 4. Fortunately, lots of your friends are of the child rearing age, so you russle up a few more babies, so that you now have 4 babies. We then tell you to divide the number of babies by 4, so you give 3 of them back to their parents, leaving you with one baby.
Now you are are told to divide the number of babies by 4. You look at the guy asking the questions really oddly, and make sure he can see how many babies are on the table. He says, "don't worry, we're going to multiply by 4 next, so you'll end up at one baby again."
For some reason, the parent who gave you the first baby really wants it back, and isn't comfortable with you having it on the table. The mother understands that, with babies (integers), 1*4/4 = 1, but 1/4*4 is NOT 1. Emphatically not 1.
This only occurred because we were using integers. Lets hand the relieved baby back to their mother, and start dividing something a little more forgiving: sticks of chewing gum.
You put one stick of chewing gum on the table. The guy asking you to do really odd things now tells you, "divide the amount of chewing gum on the table by 4. Then multiply that amount by 4. I want you to give me that much gum." Without wasting time to chop it into pieces (which you could have), you simply hand the gum over. He asks why you didn't chop anything up, so you tell him "1 stick of gum divide by four is 1/4 of a stick of gum. When I multiply 1/4 of a stick of gum by 4, I get 1 stick of gum. I'm lazy, so I just gave it to you."
Thus, for rational numbers (the integers plus the fractions), 1/4*4 = 1. Things that can be described with rational numbers can be split up in any number of ways, so its like there was a very large number of pieces, and you simply divide the pieces.
This is why mathematicians like rational numbers. They have all sorts of neat properties, like "If I name a rational number Bleep and another rational number Floop, and Floop is not the same number as zero, Bleep/Floop*Floop = Bleep." Other than making sure I didn't divide by zero, I don't actually care what Bleep and Floop were. This is really convenient for mathematicians, so we keep using rational numbers.
Disclaimer: most math with fractions is done over "real" numbers, which is these rational numbers plus another oddball set of numbers called the irrational numbers (not joking). Real numbers also have these neat properties like Bleep/Floop*Floop = Bleep, but they're a little harder to explain here. You'll find they're really intuitive to use later