0

Does $5\sqrt{5}\div5\sqrt{5}$ equal $5$ or $1$.

I think it is $1$ but I just want to check I have not missed anything.

AryanSonwatikar
  • 2,728
dagda1
  • 825
  • $(5\sqrt 5\div 5)\sqrt{5}$ or $(5\sqrt 5)\div (5\sqrt{5})$? :D – Franklin Pezzuti Dyer Nov 17 '18 at 15:34
  • 1
    Are you reading this as $(5 \sqrt 5) / (5 \sqrt 5)$ or $(5 \sqrt 5 / 5) \sqrt{5}$? This is why parentheses really matter, even if there is a standard order of operations.... Going by the usual left-to-right order where the multiplication and division have the same precedence, it's $5$. –  Nov 17 '18 at 15:35

2 Answers2

3

At face-value, it's ambiguous. But there's a convention that says evaluate from left to right, so the parens should be

$$((5\sqrt{5})\div 5)\sqrt{5} =5.$$

B. Goddard
  • 1
  • 2
    That convention works when there is an actual multiplication sign like $\cdot$ or $\times$, but I have never seen it seriously applied when it is just notated as juxtaposition. – hmakholm left over Monica Nov 17 '18 at 15:39
  • 1
    For example if you see "$\omega/2\pi$", would you expect it to mean $\frac\omega2 \pi$? – hmakholm left over Monica Nov 17 '18 at 15:47
  • @HenningMakholm I don't like the convention. It's for grade schoolers. So what I "expect" carries no weight. But grade schoolers don't divide things by $\pi.$ Adults ask for parentheses. – B. Goddard Nov 17 '18 at 15:50
3

Writing $5\cdot\sqrt5 \div 5\cdot \sqrt 5$ would be ambiguous -- it wouldn't be clear whether you mean $((5\cdot\sqrt5) \div 5)\cdot 5$ or $(5\cdot\sqrt5) \div (5\cdot \sqrt5)$.

However, when the multiplications are indicated just by placing expressions next to each other, they almost always bind tighter than operations that are notated with a visible symbol. So if someone writes $5\sqrt 5 \div 5\sqrt 5$ the probability is overwhelming that they mean $\frac{5\sqrt5}{5\sqrt 5}$, which is of course $1$.

(Or possibly they're wiseguys who are planning to select the opposite interpretation of whatever you choose. Writing $\div$ instead of $/$ or a horizontal fraction bar suggests they are not much used to mathematical conventions).

hmakholm left over Monica
  • 286,031
  • 1
    I agree. $5 \sqrt5$ can be thought of as a single (real) number, instead of a set of instructions. – M. Wind Nov 17 '18 at 17:22