User:MathMan64~enwikibooks/Arithmetic/Previous

Simple arithmetic

There are four basic operations one can perform on different numerical quantities (as well as other mathematical objects which you will encounter later). These are:

• addition
• subtraction
• multiplication
• division

There is also the use of exponents, explained briefly under Multiplication in this section and explained further in the section, Algebra/Further arithmetic, under Exponents.

It is also important to understand that the equals sign (in this book, '$=$' ) means that the things on both sides are of equal value, and that $=$ is not an operator.

When you see an $=$ sign, say to yourself Same value, different appearance.

Addition

Addition operations are denoted by the + sign. The addition operator (plus sign) will take any two numbers, called addends, as operands to work on. The result is called the sum of the two numbers.

The operation of addition is commutative. This means that the addition of two numbers will give the same sum regardless of the order in which the numbers are added.

Example:

• $7+5=12$ and $5+7=12$

• $3+2+4=9$ and $4+3+2=9$

See also Numerical axioms.

Exercises

In Exercises 1–25, find the sum.

• 1. $5+3$
• 2. $8+7$
• 3. $9+2$
• 4. $6+3$
• 5. $1+4$
• 6. $2+17$
• 7. $12+11$
• 8. $53+8$
• 9. $41+9$
• 10. $84+12$
• 11. $16+17$
• 12. $7,576+5,345$
• 13. $2,345+3,245$
• 14. $8,952+9,423$
• 15. $2,783+2,389$
• 16. $189,583+1,574,822$
• 17. $1.5+2.7$
• 18. $5.4+3.9$
• 19. $8.3+9.2$
• 20. $2.23+4.89$
• 21. $534.4+34.675$
• 22. $348.904+23,498.2$
• 23. $1.673+48,210.38$
• 24. $10.4823+94.29478$
• 25. $128.52+2,070.24$

Answers

Subtraction

Subtraction, as you probably also have seen, uses the - sign. The generic subtraction operator will take any two numbers as operands. The result is called the difference of the two numbers.

Subtraction is not a commutative operation. Changing the order of the operands will likely give a different (not the same) result.

Example:

• $7-5=2$ and $5-7=-2$

• $3-2-4=-3$ and $4-3-2=-1$

Exercises

In Exercises 26–50, find the difference.

• 26. $5-3$
• 27. $8-7$
• 28. $9-2$
• 29. $6-3$
• 30. $1-4$
• 31. $2-17$
• 32. $12-11$
• 33. $53-8$
• 34. $41-9$
• 35. $84-12$
• 36. $16-17$
• 37. $7,576-5,345$
• 38. $2,345-3,245$
• 39. $8,952-9,423$
• 40. $2,783-2,389$
• 41. $1,574,822-189,583$
• 42. $2.7-1.5$
• 43. $5.4-3.9$
• 44. $8.3-9.2$
• 45. $2.23-4.89$
• 46. $10.38-1.673$
• 47. $534.4-34.675$
• 48. $348.904-23,498.2$
• 49. $10.4823-94.29478$
• 50. $2,070.24-128.52$

Answers

Multiplication

Multiplication is denoted by a *, $\times$, or $\cdot$ sign. However, the X sign is normally not used in algebra, and is instead limited to very basic elementary math, as it can easily be confused with an "x" variable. The generic multiplication operator will take any two numbers, called factors, as operands. The result is called the product of the two numbers. If the multiplicants are not both written as numbers, the multiplication sign can be left out. Thus, the following example expressions are equivalent:

$3*\!(a*b)=3\times \!(a\times b)=3\!\cdot \!(a\!\cdot \!b)=3\!\cdot \!(ab)=3(a\!\cdot \!b)=3(ab)=3ab$

Multiplication is a form of repeated addition. For example $3\times 5$ means

$3+3+3+3+3\quad \operatorname {or} \quad 5+5+5$

Multiplication is also commutative. This means that the multiplication of two numbers (factors) will give the same product regardless of the order in which the numbers are multiplied together.

Numbers with exponents that are whole numbers larger than 1 indicate the number of factors to be multiplied, thus that number is multiplied by itself as many times as the exponent shows. Numbers with an exponent of 1 have only one factor, and therefore are equal to the number. Any number with an exponent of 0 has no factors at all, and the result is 1. Examples:

$5^{3}=5\times 5\times 5\qquad \qquad 5^{1}=5\qquad \qquad 5^{0}=1$

Further information on exponents is given under Radicals in this section and in the section, Algrebra/Further arithmetic, under Exponents.

Division

Division uses the ÷ sign. It may also be signified by the slash, /, :, or the fraction bar. The generic division operator will take any two numbers as operands. The number before the ÷ sign is called the dividend and the number after the ÷ sign is called the divisor. The result is called the quotient of the two numbers.

Division is not a commutative operation. Switching the dividend and the divisor will likely give a different quotient. The division with a divisor of 0 is not defined. There is no answer for it.

Example:

• $4/2=2$ and $2/4=0.5$

Long Division

In arithmetic, long division is an algorithm for division of two real numbers. It requires only the means to write the numbers down, and is simple to perform even for large dividends because the algorithm separates a complex division problem into smaller problems. However, the procedure requires various numbers to be divided by the divisor: this is simple with single-digit divisors, but becomes harder with larger ones.

A more generalized version of this method is used for dividing polynomials (sometimes using a shorthand version called synthetic division).

In long division notation, 500 ÷ 4 = 125 is denoted as follows:

${\begin{matrix}\quad 125\\4{\overline {)500}}\\\end{matrix}}$

The method involves several steps:

1. Write the dividend and divisor in this form:

$4{\overline {)500}}$

In this example, 500 is the dividend and 4 is the divisor.

2. Consider the leftmost digit of the dividend (5). Find the largest multiple of the divisor that is less than the leftmost digit: in other words, mentally perform "5 divided by 4". If this digit is too small, consider the first two digits.

In this case, the largest multiple of 4 that is less than 5 is 4. Write this number under the leftmost digit of the dividend. Write the multiple divided by the divisor (4 divided by 4 = 1) above the line over the leftmost digit of the dividend.

${\begin{matrix}1\\4{\overline {)500}}\\4\end{matrix}}$

3. Subtract the digit under the dividend from the digit used in the dividend. Write the result (remainder) (5 − 4 = 1) under the bottom digit, then drop the zero (the second digit) to the right of it.

${\begin{matrix}1\\4{\overline {)500}}\\{\underline {4}}\\\;\,10\end{matrix}}$

4. Repeat steps 2 and 3, except use the number you just created to divide by, and write above and under the second digit.

${\begin{matrix}\,\,12\\4{\overline {)500}}\\{\underline {4}}\\\;\,10\\\quad {\underline {8}}\\\quad \;\,20\end{matrix}}$

5. Repeat step 4 until there are no digits remaining in the dividend. The number written above the bar is the quotient, and the last remainder calculated is the remainder for the entire problem.

${\begin{matrix}\quad 125\\4{\overline {)500}}\\{\underline {4}}\\\;\,10\\\quad {\underline {8}}\\\quad \;\,20\\\quad \;\,{\underline {20}}\\\qquad 0\end{matrix}}$

Order of operations

The order of operations is the order in which all algebraic expressions should be simplified. Oftentimes, the meaning of a complex expression changes depending upon the order in which it is calculated. The order of operations is:

Parentheses

Exponents

Multiplication & Division

Addition & Subtraction

This means that expressions within parentheses are evaluated first, then exponents (including roots, i.e. radicals), then multiplication and division (at the same level), and finally addition and subtraction (at the same level). If there are multiple operations at the same level on the order of operations, move from left to right.

There are a number of different abbreviations for memorizing the order. PEMDAS, BEDMAS and BODMAS (B is Brackets) are common. Another common way to remember the order is the mnemonic

• "Please Excuse My Dear Aunt Sally,"

with the beginning letters standing for each operation. Whichever mnemonic you use, be aware that multiplication does not always come before division, and addition does not always come before subtraction. For example:

If you have an expression like

• 3 × 3 - 5 + 2

you work like this: First notice that, there are no Parenthesis or Exponents, so we move to Multiplication and Division. There's only the one multiplication, so we do that first and end up with 9 - 5 + 2. Now we move to Addition and Subtraction, working left to right. So firstly we do the subtraction to get 4 + 2, and finally the addition to give 6. If we had blindly done the addition first, we would have got the answer 2, which is wrong!

The rationale for the grouping (apart from parentheses, which are obviously first) is that multiplication is repeated addition and exponentiation is repeated multiplication. Also, division is multiplication by the reciprocal and subtraction is addition of the negative, so these operations are equivalent. In fact PEMA would be a better phrase ("Please Excuse My Aunt"), but in lower arithmetic courses MDAS is often taught without explaining reciprocals.

Parentheses are curved symbols, ( and ), that are put around part of an expression in order to convey that the expressions inside them should be evaluated first. Within a set of parentheses, the order of operations should be followed. Square brackets, [ and ], are sometimes used around parentheses to avoid confusion: [(3+5)×2]² means the same as ((3+5)×2)². The fraction bar and radical bar (often called a vinculum) groups expressions like parentheses.

For example, the expression 2×(6+7)-8² should be solved in the following order:

2×(6+7)-82	{first compute the expression inside the parentheses (6+7)}
= 2×(13)-82	{second, calculate the exponent 82}
= 2×(13)-64	{third, calculate the muliplication 2&times(13)
= 26-64	{finally, calucate the subtraction}
= -38	{our final answer}

If the desired order for solving the expression were different (based on the initial problem), parentheses would be positioned differently, or even omitted.

The meaning of the fraction and radical bars must be deciphered carefully. The part of the expression directly below or above the bar is to be treated as parenthesised. (Care must be taken in writing expressions with a bar.)

The expression ${\sqrt {a+b}}\times c$ means c times the root of a + b, not the root of a + b × c or even the root of c times the sum a + b, since the bar is above the a and the b, but not the c.

The expression ${\frac {4+5}{1+2}}$ could be written in one line as
(4+5)/(1+2) = 9/3 = 3, not as 4+5/1+2 = 4+5+2 = 11. As you see, the expression above the bar is evaluated, as is the expression below the bar. Finally we can divide.

Because of order of operations -2² = -(2²) = -4, not (-2)² = +4: the negative sign can be considered to have an implicit 0 in front, making the expression 0 - 2².

The order of operations is very important, and you must remember the order when using simple calculators. Expressions such as 2+3×5 vary on the order used. Entering [2] [+] [3] [×] [5] on most calculators would result in adding three to two and then multiplying by five, resulting in 25; the proper evaluation sequence would be 2+(3×5), multiplying three and five and then adding that to two to get 17. Some scientific calculators and most graphing calculators use the proper order of operations, but four-function calculators typically use "left-to-right" evaluation, which can return incorrect results.

See Also:

• Exponents

Fractions

A fraction consists of one quantity divided by another quantity. The fraction "three divided by five" or "three over five" or "three fifths" can be written as:

${\frac {3}{5}}$ or as: 3/5

In this section, we will use the notation 3/5. Note that some fractions have special names.

1/2 - Instead of calling this a twoth, or a second, it's called a half.

3/4 - As well as three fourths, it can also be called three quarters.

Numerator and Denominator

The first quantity, the number on top of the fraction, is called the numerator. It tells you the number, or how many of something you've got. The other number, on the bottom, is called the denominator. The denominator tells you the denomination of the fraction, which is really just a fancy way of telling you the type of the fraction. In exactly the same way, we know a £5 note and a £10 pound note are different because they are different types.

So, look at a number like 3/5. The numerator is 3. So whatever we've got, we've got three of them. The denominator is 5, so we've got fifths, whatever they are. Put the two together, we've got three of those things called fifths. We've got three fifths, we've got 3/5. Same value, different appearance.

Several rules for the calculation with fractions are useful:

Changing the type of a fraction

If both the numerator and the denominator of a fraction are multiplied or divided by the same number, then the fraction does not change its value.

This means we can make some fractions simpler, by making the numbers involved smaller. When the numbers are as small as possible, the fraction is said to be expressed in its lowest terms, or reduced. Let's look at an example:    4/6 = 2/3.

Remember what that ' = ' sign means? It means that 4/6 and 2/3, although they look different, have the same value. Same value, different appearance.

This is because

${\frac {4}{6}}={\frac {4/2}{6/2}}={\frac {2}{3}}$.

Try it out yourself with a calculator or with long division. From here, there is nothing we can divide both numbers by, so the the fraction is reduced, which is our goal.

Adding fractions

Before we go into fractions, let's have a think about what addition is. Answer these really simple questions.

• What is 1 bird + 5 birds?
• What is 3 elephants + 9 elephants?
• What is 6 birds + 2 elephants?

Hopefully you could answer the first two, but you might have had trouble with the third. Why? Because we can't add up two different things! However, before we add them up, we can change them into something else. We can say that 6 birds are really just 6 animals, and 2 elephants are really 2 animals, so now

6 birds + 2 elephants = 8 animals.

You probably did all this in your head without even thinking about it - so what has it all got to do with fractions?

To add or subtract two fractions, you first need to change the two fractions so that they have the same type. The simplest way to do this is to multiply the numerator and denominator of each fraction by the denominator of the other.

For instance,

${\frac {2}{3}}+{\frac {1}{4}}={\frac {2\times 4}{3\times 4}}+{\frac {1\times 3}{4\times 3}}={\frac {8}{12}}+{\frac {3}{12}}={\frac {(8+3)}{12}}={\frac {11}{12}}$

A more advanced way is to use the LCM of the denominators, which will be explained later in this section. Then you can add or subtract the numerators and put the common denominator as the denominator of the solution.

Multiplying fractions

To multiply two fractions:

• multiply the numerators to get the new numerator, and
• multiply the denominators to get the new denominator.

For instance,

${\frac {2}{3}}\times {\frac {1}{4}}={\frac {2\times 1}{3\times 4}}={\frac {2}{12}}={\frac {1}{6}}$.

Dividing fractions

To divide one fraction by another one, flip numerator and denominator of the second one, and then multiply the two fractions. The flipped-over fraction is called the multiplicative inverse or reciprocal.

For instance,

$\left({\frac {2}{3}}\right)/\left({\frac {4}{5}}\right)={\frac {2}{3}}\times {\frac {5}{4}}={\frac {2\times 5}{3\times 4}}={\frac {10}{12}}={\frac {5}{6}}$.

To simplify a compound fraction, like ${\frac {\left({\frac {3}{5}}\right)}{\left({\frac {1}{4}}\right)}}$, just remember that a fraction is the same as division, and divide (3/5) ÷ (1/4), which comes to 12/5.

Mixed fractions

Sometimes you may meet 'mixed fractions' like $1{\frac {3}{5}}$. This really means $1+{\frac {3}{5}}$, or in words one and three-fifths. Mixed fractions are hard to work with... how would you go about working out the following?

$1{\frac {3}{5}}\times 2{\frac {2}{7}}$

The answer is to convert the mixed fraction into a standard fraction of the form numerator/denominator. For example,

$1{\frac {3}{5}}=1+{\frac {3}{5}}={\frac {5}{5}}+{\frac {3}{5}}={\frac {8}{5}}\,\,\,\,$

and

$2{\frac {2}{7}}=2+{\frac {2}{7}}={\frac {14}{7}}+{\frac {2}{7}}={\frac {16}{7}}$

Now, it's simple to multiply the fractions together:

$1{\frac {3}{5}}\times 2{\frac {2}{7}}={\frac {8}{5}}\times {\frac {16}{7}}={\frac {128}{35}}={\frac {105}{35}}+{\frac {23}{35}}=3{\frac {23}{35}}$

External links

• Online interactive exercises on fractions.

Absolute value

The absolute value of a number is found by applying a simple rule: If you see a negative sign in front of the number, change it to a plus sign. If you see a plus sign, leave it alone. So, for example, the absolute value of -17 is +17. The absolute value of +36 is +36.

Another way to understand the absolute value of a number is to think about the number line:

The absolute value of a number is the distance from zero to that number on the number line.

The absolute value of x is usually written as |x|. On calculators and computers it is sometimes written as abs (x).

Questions:

1. Calculate the absolute value of the following numbers:

a. -5

b. 9

c. -3.8

d. -139,462

e. 5/8

2. What is the absolute value of zero? Explain.

3. Calculate the following:

a. |27|

b. |-1.9|

c. |3 - 7|

d. |3 - 0.5|

e. abs (-6)

4. Draw a graph of abs(x) from -5 to +5. Can abs(x) ever be less than zero? How can you see that from your graph?

Answers:

1. a. 5 b. 9 c. 3.8 d. 139,462 e. 5/8

2. Zero, because zero is exactly zero away from zero on the number line.

3. a. 27 b. 1.9 c. |3-7| = |-4| = 4 d. |3-0.5| = |2.5| = 2.5 e. 6

4. Image of the absolute value of x

Radicals

A radical is a special kind of number that is the root of a polynomial equation. First, let us look at one specific kind of radical, the "square root". It is a special kind of number related to squaring.

When we have a number, say 2, what positive number will give us, when we square it, the number 2?

• 1 × 1 = 1, so the number must be higher than 1.
• 2 × 2 = 4, so the number must be lower than 2.
• 1.5 × 1.5 = 2.25, so the number must be lower than 1.5.

We could continue like this forever, and with each step get closer and closer to the answers (this kind of process for refining the answer is called iteration). The number we are looking for is approximately 1.41421...

Obviously this is very difficult for us to work with, so we have a special notation. We write for a number a, ${\sqrt {a}}$ to represent the number when squared will give us a back.

Since this is the inverse operation from squaring, it can also be denoted as a^1/2 and
$(a^{1/2})^{2}=a^{1}=a$.

We can extend this idea to other kinds of radicals. ${\sqrt[{3}]{a}}$ indicates the number $x$ such that
$x^{3}=a$. For example, ${\sqrt[{3}]{7}}=1.91293...$

Note that it is not possible to find any real number whose square would be negative. Multiplying with a negative number changes the sign of the number being multiplied and two negative signs thus eliminate each other. For example, -7 × -7 = 49 and also 7 × 7 = 49 . Therefore the square root of a negative number is an undefined operation unless imaginary numbers are allowed as answers.

Estimation

Estimation involves working out a rough answer to a calculation. The most common way to estimate a solution to a calculation is to round the numbers up or down to numbers which are easier to calculate with.

An example: Estimate the answer to 99 × 9. In this case we can easily see that 9 is almost 10, and multiplying by 10 is really easy, so we can replace our calculation with 99 × 10 which is far more easy to calculate. As we rounded a number up we know that our estimate is going to be larger than the real answer.

Why don't we just call this a guess? The difference is that a guess is just that, a completely wild guess. An estimate is based on some extra information. So whilst you might guess that
99 × 9 is something around 1000 by just pulling a number out of the air, we estimate that 99 × 9 is close to 99 × 10, then we work out 99 × 10 exactly, which gives 990 as an estimate of
99 × 9.

When we estimate, we want our estimate to be close enough to the actual value so as to be useful. (This also applies to approximation.) In our example above, the answer deviates from the actual value by about 10%, which might not be acceptable. Estimation often involves some guesswork, so we might not actually know how accurate our estimate is. (And yes, that figure 10% is itself an estimate.)

A better way to estimate 99 × 9 is to say that 99 is close to 100, then we work out 100 × 9 exactly, which gives 900 as an estimate of 99 × 9. This estimate is about 1% off. The reason this is a better estimate is that 99 deviates from 100 much less than 9 deviates from 10 in percentage terms (although the absolute difference is one in each case).