#### • Class 11 Physics Demo

Explore Related Concepts

# higher order thinking questions math

From Wikipedia

Order of operations

In mathematics and computer programming, the order of operations (more formally precedence rule) is a rule used to unambiguously clarify which procedures should be performed first in a given mathematical expression.

For example, in mathematics and most computer languages multiplication is done first; in the expression 2 + 3 Ã— 4, the algebraic answer is 14. Parentheses, which have their own rules, may be used to avoid confusion, thus the above expression may also be rendered 2 + (3 Ã— 4).

From the introduction of modern algebraic notation, where juxtaposition indicates multiplication of variables, multiplication took precedence over addition, whichever side of a number it appeared on. Thus 3&nbsp;+&nbsp;4&nbsp;&times;&nbsp;5 =&nbsp;4&nbsp;&times;&nbsp;5&nbsp;+&nbsp;3 = 23. When exponents were first introduced, in the 16th and 17th centuries, exponents took precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base. Thus 3&nbsp;+&nbsp;52 = 28 and 3&nbsp;&times;&nbsp;52 =&nbsp;75. To change the order of operations, originally a vinculum (an overline or underline) was used. Today we use parentheses. Thus, to force addition to precede multiplication, we write (2&nbsp;+&nbsp;3)&nbsp;&times;&nbsp;4 =&nbsp;20, and to force addition to precede exponentiation, we write (3&nbsp;+&nbsp;5)2 = 64

## The standard order of operations

The standard order of operations, or precedence, is expressed in the following chart.

terms inside brackets
exponents and roots
multiplication and division

This means that if a number or other symbol, or an expression grouped by one or more symbols of grouping, is preceded by one operator and followed by another, the operator higher on the list should be applied first. The commutative and associative laws of addition and multiplication allow the operators +, âˆ’, *, and / to be applied in any order that obeys this rule. The root symbol, âˆš, requires either parentheses around the radicand or a bar (called vinculum) over the radicand. Stacked exponents are applied from the top down.

It is helpful to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the opposite (additive inverse). Thus 3/4 =&nbsp;3&nbsp;Ã·&nbsp;4 = 3&nbsp;â€¢&nbsp;Â¼ and 3&nbsp;âˆ’&nbsp;4 = 3&nbsp;+&nbsp;(âˆ’4), that is, the sum of positive three and negative four.

Symbols of grouping can be used to override the usual order of operations. Grouped symbols can be treated as a single expression. Symbols of grouping can be removed using the associative and distributive laws.

### Examples

\sqrt{1+3}+5=\sqrt4+5=2+5=7.\,

A horizontal fractional line also acts as a symbol of grouping:

\frac{1+2}{3+4}+5=\frac37+5.

For ease in reading, other grouping symbols (such as curly braces {} or square brackets [] ) are often used along with the standard round parentheses, e.g.

[(1+2)-3]-(4-5) = [3-3]-(-1) = 1. \,

Unfortunately, there exist differing conventions concerning the unary operator âˆ’ (usually read "minus"). In written or printed mathematics, the expression &minus;32 is interpreted to mean &minus;(32)&nbsp;=&nbsp;&minus;9, but in some applications and programming languages, notably the application Microsoft Office Excel and the programming language bc, unary operators have a higher priority than binary operators, that is, the unary minus (negation) has higher precedence than exponentiation, so in those languages &minus;32 will be interpreted as (&minus;3)2&nbsp;=&nbsp;9. [http://support.microsoft.com/kb/q132686/]. In any case where there is a possibility that the notation might be misinterpreted, it is advisable to use parentheses to clarify which interpretation is intended.

Similarly, care must be exercised when using the slash ('/') symbol. The string of characters&nbsp;"1/2x" is interpreted by the above conventions as&nbsp;(1/2)x. The contrary interpretation should be written explicitly as 1/(2x). Again, the use of parentheses will clarify the meaning and should be used if there is any chance of misinterpretation.

Mnemonics are often used to help students remember the rules, but the rules taught by the use of acronyms can be misleading. In the United States, the acronymPEMDAS or "Please Excuse My Dear Aunt Sally" is common. It stands for Parentheses, Exponentiation, Multiplication, Division, Addition, Subtraction. In other English speaking countries, Parentheses may be called Brackets, and Exponentiation may be called either Indices, Powers or Orders, and since multiplication and division are of equal precedence, M and D are often interchanged, leading to such acronyms as BEDMAS, BIDMAS, BIMDAS, BODMAS, BOMDAS, BERDMAS, PERDMAS, and BPODMAS.

These mnemonics may be misleading, especially if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction. Using any of the above rules in the order "addition first, subtraction afterward" would also give the wrong answer.

10 - 3 + 2 \,

The correct answer is 9, which is best understood by thinking of the problem as the sum of positive ten, negative three, and positive two.

10 + (-3) + 2 \,

There is a new mnemonic featured in Danica McKellar's books Math Doesn't Suck and Kiss My Math that does address this very issue: "Pandas Eat: Mustard on Dumplings, and Apples with Spice." The intention being that Mustard and Dumplings is a "dinner course" and that Apples and Spice is a "dessert course." Then it becomes not a linear string of operations to do one after the other, but rather the "dinner course" operations are considered together and performed left to right, and then addition and subtraction are considered together, again performed again left to right.

In college mathematics, the rules of priority are (usually) taught correctly, and students are taught the commutative law, associative law, and distributive law, which replace the grade school "rules". The "left to right" rule is not a law of mathematics.

For example,

17 \times 24 / 12 \,

is much easier whe

Question:express a-b/a+b - a+b/a+2b showing all workings what is stratified random sampling? and what is an advantage of stratified random sampling?

Answers:Given a-b/a+b - a+b/a+2b do you really mean [a-b] /[a+b] - [a+b] / [a+2b] ? If so then you really need to use brackets. What you have written will be computed as a - [b/a] + b - a + [b] / [a+2b ] EMPHASISE what you mean and resubmit

Question:It is the british scholar book, exercise 7 question 51. i have been given the answer but i dont know how to do it step by step, help would be greatly appreciated. the question is : integrate with limits 0, to 1/4 (the funny little integrate sign with the numbers at top and bottom) then it is SQUARE ROOT OF (1-4x^2) dx. and it says using x= 1/2 sin T we're doing integration with definite integrals and with substitution and this is a difficult one.

Answers:(1 - 4x^2) dx from [ 0 to 1/4 ] let 2x = sin(t) ==> 4x^2 = sin^2(t) x = (1/2) sin(t) , when x = 0, t = 0 and when x = 1/4, t = /6 dx = (1/2) cos(t) dt now the integral becomes (1 - sin^2(t)) (1/2) cos(t) dt from [ 0 to /6 ] = 1/2 cos^2(t) cos(t) dt from [ 0 to /6 ] = 1/2 cos^2(t) dt from [ 0 to /6 ] = 1/2 1/2 [ 1 + cos(2t) ] dt from [ 0 to /6 ] = 1/4 [ 1 + cos(2t) ] dt from [ 0 to /6 ] = (1/4) t + (1/8) sin(2t) from [ 0 to /6 ] = (1/4) [ /6 - 0 ] + 1/8 [ sin( /3) - sin(0) ] = (1/24) + (1/8) 3/2 = (1/24) + (1/16) 3