# Real life example to explain the Difference between Algebra and Arithmetic [closed]

Can anyone help me in finding out some real life examples to explain the difference between algebra and arithmetic ?

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## closed as not a real question by Pedro Tamaroff♦, Andrés Caicedo, Austin Mohr, Chris Eagle, WilliamAug 15 '12 at 16:28

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No you don't use anything :) – timur Aug 14 '12 at 23:53
What is arithmetic? Is it computing $1+1=$? Algebra would be figuring out how to find $x$ from $x-1=1$? – timur Aug 15 '12 at 0:04
Both "algebra" and "arithmetic" are highly overloaded terms. To properly answer your question, we need to know precisely what you mean by those terms. How much mathematics do you know? – Bill Dubuque Aug 15 '12 at 0:22
@Hillbilli: Two questions have been asked of you (by Bill, J.M., and I) that you should answer: What do you mean by algebra and arithmetic? How much math do you know? If you continue to brush these questions aside, we will not be able to give you a good answer. – mixedmath Aug 15 '12 at 0:44
@Hillbilli: I have now downvoted, and I wonder if this is a troll. I will be happy to remove my downvote if you explain what you mean by algebra and arithmetic, and explain the intended level of math. – mixedmath Aug 15 '12 at 1:41

A very famous book about algebra, the one that gave algebra its very name, was written around 1200 years ago by a Muslim mathematician called al-Khwarizmi. One of the types of real-life problems it treats extensively is that of calculating inheritances under the Islamic inheritance law of the time.

Here is one example from that book:

A woman dies, leaving her husband, a son, and three daughters.

The implied question is: how is the woman's estate to be divided among the five heirs?

Under the law of the time, the husband is entitled to ¼ of the woman's estate, daughters to equal shares of the remainder, and sons to shares that are twice the daughters'. So a little arithmetic will suffice to solve this simple problem: We begin with the husband, who gets 25%. The son gets two shares and three daughters each get one share of the remaining 75%. That totals five shares, so each share is 75% ÷ 5 = 15%, and thus the son gets 30% and the daughters 15% each. No algebra was necessary, just direct calculation.

But a more complicated problem, also treated by al-Khwarizmi, goes like this:

A man dies, leaving two sons and bequeathing one-third of his estate to a stranger. His estate consists of ten dirhams of ready cash and ten dirhams as a claim against one of the sons, to whom he has loaned the money.

The law in this case says that the stranger gets to collect his legacy before the shares of the rest of the estate is computed, and that if the debtor son's share of the estate is not large enough to enable him to pay back the debt completely, the remainder is written off as uncollectable.

Here arithmetic is not enough, because there is nowhere to begin. We need to know the debtor son's share to calculate the amount of the writeoff. But we need to know the writeoff to calculate the total value of the estate, we need the value of the estate to calculate the bequest to the stranger, and we need to know the size of the bequest to calculate the size of the sons' shares.

The circularity makes this a problem in algebra rather than arithmetic, and we must solve it using algebraic technique: Let $e$ be the total value of the estate, $c$ be the amount of cash, $d$ be the debtor son's debt, $w$ the amount of the writeoff, $b$ be the bequest to the stranger, and $s$ be each son's share of the estate. Then we have:

$$\begin{eqnarray} c & = & 10 \\ d & = & 10 \\ e & = & c + (d - w) \\ w & = & d - s \\ b & = & e / 3 \\ s & = & (e - b) / 2 \end{eqnarray}$$

We can solve these and find $w=5$, $e=15$, $b=5$, $s=5$, so 5 dirhams of the debt is written off, leaving an estate of 15 dirhams. The stranger gets one-third of this amount, 5 dirhams in cash, the non-debtor son gets the other 5 dirhams cash, and the debtor son's share is to have the remaining 5 dirhams of his debt forgiven.

I took these problems from Episodes in the Mathematics of Medieval Islam, by J.L. Berggren (Springer, 1983).

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So, elementary arithmetic is about the basic operations in maths that teach people how to combine the known quantities to derive the desired result. – FrenzY DT. Aug 15 '12 at 3:25
youtube.com/watch?v=8JSG7AMGilM&feature=related...This is the link that also explains my question – Hillbilli Aug 15 '12 at 4:55
Hillbilli, the place to explain your question is in your question, not in a comment on an answer. And a self-contained explanation trumps a call-out to YouTube. – Gerry Myerson Aug 15 '12 at 5:02
@Hillbilli I'm not going to watch the video, but if you edit your question to explain further what you want to know, I'll try to answer. – MJD Aug 15 '12 at 12:23
Shouldn't the write-off be $w=\max\{0,d-s\}$? – timur Aug 15 '12 at 14:35

If you go the 20 miles from Pleasantville to Happytown at 20 miles per hour, and make the return journey at 10 miles per hour, what's your average speed? That's arithmetic.

If you go the 20 miles at 20 miles per hour, how fast do you have to make the return journey to have an average speed of 40 miles per hour? That's algebra.

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+1. I think this answer is much more apt than mixedmath's high-powered one. – Ganesh Aug 15 '12 at 2:40

I'd like to give you an example of why we wanted clarification about what you meant by algebra and arithmetic, or the intended level of audience. So this is taking the opposite interpretation as Gerry, deliberately.

When I think of arithmetic, I think of my recent study of A Course in Arithmetic by Serre. This serves as an excellent introduction to the idea of modular forms. What might be a real-world application of modular forms? Why, black holes, of course. A little bit more down-to earth, we might mention that there are cryptosystems that rely on modular forms.

When I think of algebra, I think of Galois Theory or algebraic number theory. A key example application of Galois Theory is that if we have a general quintic $ax^5 + bx^4 + cx^3 + dx^2 + ed + f = 0$, we can't factor it with just addition, multiplication, and radicals. We care about factoring polynomials, just like we care about factoring numbers. I mention this because one of the most important and fundamental results of algebraic number theory is the Fundamental Theorem of Arithmetic. That's a bit of irony, right?

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Here is a different kind of answer, which may or may not help, given the ambiguity of the question and the deep philosophical questions it raises (like "what is a number").

Arithmetic could roughly be described as working with the numbers we know within a particular system of numbers, and is often related in some way to working with things called integers (whole numbers) and fractions. These ideas are so useful that they've been generalised and abstracted by mathematicians many times - so there are 'integral domains' and 'rings of integers' and 'fields of fractions' and more to be learned about in due course.

Algebra might be roughly characterised as analysing the properties of mathematical systems which are in some way like numbers (things you can add or multiply) - though there are things there which we don't always think of as numbers like vector spaces and groups.

Roughly speaking we can use algebra to solve problems in arithmetic by putting our specific arithmetic problems in a more general algebraic context which gives more insight or options. So in elementary work we allow ourselves to use $x$ as an unknown number having the same basic properties as the numbers we know, and manipulate it as if it is a number until we find out what it must be. Solving polynomial equations we change our context to a 'splitting field' for the equation and analyse its properties, which gives us information about the roots.

This does not do proper justice to the terms, I think, because the distinction between algebra and arithmetic cannot be clearly made in this way (As others have suggested), but I hope it clarifies some ideas rather than clouding them.

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