# Real Numbers

### Learning Outcomes

- Add and subtract real numbers
- Add real numbers with the same and different signs
- Subtract real numbers with the same and different signs
- Simplify combinations that require both addition and subtraction of real numbers.

- Multiply and divide real numbers
- Multiply two or more real numbers.
- Divide real numbers
- Simplify expressions with both multiplication and division
- Simplify expressions with real numbers

- Recognize and combine like terms in an expression
- Simplify compound expressions with real numbers
- Simplify expressions with fraction bars, brackets, and parentheses
- Use the distributive property to simplify expressions with grouping symbols
- Simplify expressions containing absolute values

**integers:**counting numbers like 1, 2, 3, etc., including negatives and zero**real number:**fractions, negative numbers, decimals, integers, and zero are all real numbers**absolute value:**a number's distance from zero; it's always positive. [latex]|-7| = 7[/latex]**sign:**this refers to whether a number is positive or negative, we use [latex]+[/latex] for positive (to the right of zero on the number line) and [latex]-[/latex] for negative (to the left of zero on the number line)**difference:**the result of subtraction**sum:**the result of addition

## Adding and Subtracting Real Numbers

When adding integers we have two cases to consider. The first case is whether the signs match (both positive or both negative). If the signs match, we will add the numbers together and keep the sign. If the signs don’t match (one positive and one negative number) we will subtract the numbers (as if they were all positive) and then use the sign from the larger number. This means if the larger number is positive, the answer is positive. If the larger number is negative, the answer is negative.### To add two numbers with the same sign (both positive or both negative)

*Add*their absolute values (without the [latex]+[/latex] or [latex]-[/latex] sign)- Give the sum the same sign.

### To add two numbers with different signs (one positive and one negative)

- Find the
*difference*of - Give the sum the same sign as the number with the greater absolute value.

### Example

Find [latex]23–73[/latex].Answer: You can't use your usual method of subtraction because 73 is greater than 23. Rewrite the subtraction as adding the opposite.

[latex]23+\left(−73\right)[/latex]

The addends have different signs, so find the difference of their absolute values.[latex]\begin{array}{c}\left|23\right|=23\,\,\,\text{and}\,\,\,\left|−73\right|=73\\73-23=50\end{array}[/latex]

Since [latex]\left|−73\right|>\left|23\right|[/latex], the final answer is negative.#### Answer

[latex-display]23–73=−50[/latex-display]*right*of 0 by [latex]382[/latex] units, and [latex]−93[/latex] is to the

*left*of 0 by 93 units. The distance between them is the sum of their distances to 0: [latex]382+93[/latex].

### Example

Find [latex]382–\left(−93\right)[/latex].Answer: You are subtracting a negative, so think of this as taking the negative sign away. This becomes an addition problem. [latex]-93[/latex] becomes [latex]+93[/latex]

[latex]382+93=475[/latex]

#### Answer

[latex-display]382–(−93)=475[/latex-display]### Example

Find [latex]-\frac{3}{7}-\frac{6}{7}+\frac{2}{7}[/latex]Answer: Add the first two and give the result a negative sign: Since the signs of the first two are the same, find the sum of the absolute values of the fractions Since both numbers are negative, the sum is negative. If you owe money, then borrow more, the amount you owe becomes larger.

[latex]\left| -\frac{3}{7} \right|=\frac{3}{7}[/latex] and [latex]\left| -\frac{6}{7} \right|=\frac{6}{7}[/latex]

[latex]\begin{array}{c}\frac{3}{7}+\frac{6}{7}=\frac{9}{7}\\\\-\frac{3}{7}-\frac{6}{7} =-\frac{9}{7}\end{array}[/latex]

Now add the third number. The signs are different, so find the*difference*of their absolute values.

[latex] \left| -\frac{9}{7} \right|=\frac{9}{7}[/latex] and [latex] \left| \frac{2}{7} \right|=\frac{2}{7}[/latex]

[latex] \frac{9}{7}-\frac{2}{7}=\frac{7}{7}[/latex]

Since [latex]\left|\frac{-9}{7}\right|>\left|\frac{2}{7}\right|[/latex], the sign of the final sum is the same as the sign of [latex]-\frac{9}{7}[/latex].

[latex] -\frac{9}{7}+\frac{2}{7}=-\frac{7}{7}[/latex]

#### Answer

[latex-display]-\frac{3}{7}+\left(-\frac{6}{7}\right)+\frac{2}{7}=-\frac{7}{7}[/latex-display]### Example

Evaluate [latex]27.832+(−3.06)[/latex]. When you add decimals, remember to line up the decimal points so you are adding tenths to tenths, hundredths to hundredths, and so on.Answer: Since the addends have different signs, subtract their absolute values.

[latex] \begin{array}{r}\underline{\begin{array}{r}27.832\\-\text{ }3.06\,\,\,\end{array}}\\24.772\end{array}[/latex]

[latex]\left|-3.06\right|=3.06[/latex]

The sum has the same sign as 27.832 whose absolute value is greater.#### Answer

[latex-display]27.832+\left(-3.06\right)=24.772[/latex-display]## Multiplying and Dividing Real Numbers

Multiplication and division are**inverse operations**, just as addition and subtraction are. You may recall that when you divide fractions, you multiply by the reciprocal. Inverse operations "undo" each other.

## Multiply Real Numbers

Multiplying real numbers is not that different from multiplying whole numbers and positive fractions. However, you haven't learned what effect a negative sign has on the product. With whole numbers, you can think of multiplication as repeated addition. Using the number line, you can make multiple jumps of a given size. For example, the following picture shows the product [latex]3\cdot4[/latex] as 3 jumps of 4 units each. So to multiply [latex]3(−4)[/latex], you can face left (toward the negative side) and make three “jumps” forward (in a negative direction). The product of a positive number and a negative number (or a negative and a positive) is negative.### The Product of a Positive Number and a Negative Number

To multiply a**positive number**and a

**negative number**, multiply their absolute values. The product is

**negative**.

### Example

Find [latex]−3.8(0.6)[/latex].Answer: Multiply the absolute values as you normally would. Place the decimal point by counting place values. 3.8 has 1 place after the decimal point, and 0.6 has 1 place after the decimal point, so the product has [latex]1+1[/latex] or 2 places after the decimal point.

[latex]\begin{array}{r}3.8\\\underline{\times\,\,\,0.6}\\2.28\end{array}[/latex]

The product of a negative and a positive is negative.#### Answer

[latex-display]−3.8(0.6)=−2.28[/latex-display]### The Product of Two Numbers with the Same Sign (both positive or both negative)

To multiply two**positive numbers**, multiply their absolute values. The product is

**positive**. To multiply two

**negative numbers**, multiply their absolute values. The product is

**positive**.

### Example

Find [latex] ~\left( -\frac{3}{4} \right)\left( -\frac{2}{5} \right)[/latex]Answer: Multiply the absolute values of the numbers. First, multiply the numerators together to get the product's numerator. Then, multiply the denominators together to get the product's denominator. Rewrite in lowest terms, if needed.

[latex] \left( \frac{3}{4} \right)\left( \frac{2}{5} \right)=\frac{6}{20}=\frac{3}{10}[/latex]

The product of two negative numbers is positive.#### Answer

[latex-display] \left( -\frac{3}{4} \right)\left( -\frac{2}{5} \right)=\frac{3}{10}[/latex-display]**positive**[latex]\cdot[/latex]The product is*positive*:**positive**.**negative**[latex]\cdot[/latex]The product is*negative*:**positive**.**negative**[latex]\cdot[/latex]The product is*positive*:**negative**.**positive**[latex]\cdot[/latex]The product is*negative*:**negative**.

### Multiplying More Than Two Negative Numbers

If there are an**even**number (0, 2, 4, ...) of negative factors to multiply, the product is

**positive**. If there are an

**odd**number (1, 3, 5, ...) of negative factors, the product is

**negative**.

### Example

Find [latex]3(−6)(2)(−3)(−1)[/latex].Answer: Multiply the absolute values of the numbers.

[latex]\begin{array}{l}3(6)(2)(3)(1)\\18(2)(3)(1)\\36(3)(1)\\108(1)\\108\end{array}[/latex]

Count the number of negative factors. There are three [latex]\left(−6,−3,−1\right)[/latex].[latex]3(−6)(2)(−3)(−1)[/latex]

Since there are an odd number of negative factors, the product is negative.#### Answer

[latex-display]3(−6)(2)(−3)(−1)=−108[/latex-display]## Divide Real Numbers

You may remember that when you divided fractions, you multiplied by the**reciprocal**.

*Reciprocal*is another name for the multiplicative inverse (just as

*opposite*is another name for additive inverse). An easy way to find the multiplicative inverse is to just “flip” the numerator and denominator as you did to find the reciprocal. Here are some examples:

- The reciprocal of [latex]\frac{4}{9}[/latex] is [latex] \frac{9}{4}[/latex]because [latex]\frac{4}{9}\left(\frac{9}{4}\right)=\frac{36}{36}=1[/latex].
- The reciprocal of 3 is [latex]\frac{1}{3}[/latex] because [latex]\frac{3}{1}\left(\frac{1}{3}\right)=\frac{3}{3}=1[/latex].
- The reciprocal of [latex]-\frac{5}{6}[/latex] is [latex]\frac{-6}{5}[/latex] because [latex]-\frac{5}{6}\left( -\frac{6}{5} \right)=\frac{30}{30}=1[/latex].
- The reciprocal of 1 is 1 as [latex]1(1)=1[/latex].

**real numbers**. Think about dividing a bag of 26 marbles into two smaller bags with the same number of marbles in each. You can also say each smaller bag has

*one half*of the marbles.

[latex] 26\div 2=26\left( \frac{1}{2} \right)=13[/latex]

Notice that 2 and [latex] \frac{1}{2}[/latex] are reciprocals. Try again, dividing a bag of 36 marbles into smaller bags.Number of bags | Dividing by number of bags | Multiplying by reciprocal |
---|---|---|

3 | [latex]\frac{36}{3}=12[/latex] | [latex] 36\left( \frac{1}{3} \right)=\frac{36}{3}=\frac{12(3)}{3}=12[/latex] |

4 | [latex]\frac{36}{4}=9[/latex] | [latex]36\left(\frac{1}{4}\right)=\frac{36}{4}=\frac{9\left(4\right)}{4}=9[/latex] |

6 | [latex]\frac{36}{6}=6[/latex] | [latex]36\left(\frac{1}{6}\right)=\frac{36}{6}=\frac{6\left(6\right)}{6}=6[/latex] |

**divisor**, the second number in the division problem.)

### Example

Find [latex] 28\div \frac{4}{3}[/latex]Answer: Rewrite the division as multiplication by the reciprocal. The reciprocal of [latex] \frac{4}{3}[/latex] is [latex]\frac{3}{4} [/latex].

[latex] 28\div \frac{4}{3}=28\left( \frac{3}{4} \right)[/latex]

Multiply.[latex]\frac{28}{1}\left(\frac{3}{4}\right)=\frac{28\left(3\right)}{4}=\frac{4\left(7\right)\left(3\right)}{4}=7\left(3\right)=21[/latex]

#### Answer

[latex-display]28\div\frac{4}{3}=21[/latex-display]### Rules of Division

When dividing, rewrite the problem as multiplication using the reciprocal of the divisor as the second factor. When one number is**positive**and the other is

**negative**, the

**quotient**is

**negative**. When

*both*numbers are

**negative**, the quotient is

**positive**. When

*both*numbers are

**positive**, the quotient is

**positive**.

### Example

Find [latex]24\div\left(-\frac{5}{6}\right)[/latex].Answer: Rewrite the division as multiplication by the reciprocal.

[latex] 24\div \left( -\frac{5}{6} \right)=24\left( -\frac{6}{5} \right)[/latex]

Multiply. Since one number is positive and one is negative, the product is negative.[latex] \frac{24}{1}\left( -\frac{6}{5} \right)=-\frac{144}{5}[/latex]

#### Answer

[latex-display] 24\div \left( -\frac{5}{6} \right)=-\frac{144}{5}[/latex-display]### Example

Find [latex] 4\,\left( -\frac{2}{3} \right)\,\div \left( -6 \right)[/latex]Answer: Rewrite the division as multiplication by the reciprocal.

[latex] \frac{4}{1}\left( -\frac{2}{3} \right)\left( -\frac{1}{6} \right)[/latex]

Multiply. There is an even number of negative numbers, so the product is positive.[latex]\frac{4\left(2\right)\left(1\right)}{3\left(6\right)}=\frac{8}{18}[/latex]

Write the fraction in lowest terms.#### Answer

[latex-display] 4\left( -\frac{2}{3} \right)\div \left( -6 \right)=\frac{4}{9}[/latex-display]## Simplify Expressions With Real Numbers

Some important terminology before we begin:**operations/operators:**In mathematics we call things like multiplication, division, addition, and subtraction operations. They are the verbs of the math world, doing work on numbers and variables. The symbols used to denote operations are called operators, such as [latex]+{, }-{, }\times{, }\div[/latex]. As you learn more math, you will learn more operators.**term:**Examples of terms would be [latex]2x[/latex] and [latex]-\frac{3}{2}[/latex] or [latex]a^3[/latex]. Even lone integers can be a term, like 0.**expression:**A mathematical expression is one that connects terms with mathematical operators. For example [latex]\frac{1}{2}+\left(2^2\right)- 9\div\frac{6}{7}[/latex] is an expression.

## Combining Like Terms

One way we can simplify expressions is to combine like terms.**Like terms**are terms where the variables match exactly (exponents included). Examples of like terms would be [latex]5xy[/latex] and [latex]-3xy[/latex] or [latex]8a^2b[/latex] and [latex]a^2b[/latex] or [latex]-3[/latex] and [latex]8[/latex]. If we have like terms we are allowed to add (or subtract) the numbers in front of the variables, then keep the variables the same. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term. This is shown in the following examples:

### Example

Combine like terms: [latex]5x-2y-8x+7y[/latex]Answer: The like terms in this expression are:

[latex]5x[/latex] and [latex]-8x[/latex]

[latex]-2y[/latex] and [latex]-7y[/latex]

Note how we kept the sign in front of each term.

Combine like terms:

[latex]5x-8x = -3x[/latex]

[latex]-2y-7y = -9y[/latex]

Note how signs become operations when you combine like terms.

Simplified Expression:

[latex]5x-2y-8x+7y=-3x-9y[/latex]

### Example

Combine like terms: [latex]x^2-3x+9-5x^2+3x-1[/latex]Answer: The like terms in this expression are:

[latex]x^2[/latex] and [latex]-5x^2[/latex]

[latex]-3x[/latex] and [latex]3x[/latex]

[latex]9[/latex] and [latex]-1[/latex]

Combine like terms:

[latex]\begin{array}{r}x^2-5x^2 = -4x^2\\-3x+3x=0\,\,\,\,\,\,\,\,\,\,\,\\9-1=8\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

## Simplify Compound Expressions With Real Numbers

In this section, we will use the skills from the last section to simplify mathematical expressions that contain many grouping symbols and many operations. We are using the term compound to describe expressions that have many operations and many grouping symbols. More care is needed with these expressions when you apply the order of operations. Additionally, you will see how to handle absolute value terms when you simplify expressions.### Example

Simplify [latex] \frac{5-[3+(2\cdot (-6))]}{{{3}^{2}}+2}[/latex]Answer: This problem has brackets, parentheses, fractions, exponents, multiplication, subtraction, and addition in it. Grouping symbols are handled first. The parentheses around the [latex]-6[/latex] aren’t a grouping symbol; they are simply making it clear that the negative sign belongs to the 6. Start with the innermost set of parentheses that are a grouping symbol. In this example, the innermost set of parentheses would be in the numerator of the fraction, [latex](2\cdot(−6))[/latex]. Begin working out from there. (The fraction line acts as a type of grouping symbol, too; you simplify the numerator and denominator independently, and then divide the numerator by the denominator at the end.)

[latex]\begin{array}{c}\frac{5-\left[3+\left(2\cdot\left(-6\right)\right)\right]}{3^{2}+2}\\\\\frac{5-\left[3+\left(-12\right)\right]}{3^{2}+2}\end{array}[/latex]

Add [latex]3[/latex] and [latex]-12[/latex], which are in brackets, to get [latex]-9[/latex].[latex]\begin{array}{c}\frac{5-\left[3+\left(-12\right)\right]}{3^{2}+2}\\\\\frac{5-\left[-9\right]}{3^{2}+2}\end{array}[/latex]

Subtract [latex]5–\left[−9\right]=5+9=14[/latex].[latex]\begin{array}{c}\frac{5-\left[-9\right]}{3^{2}+2}\\\\\frac{14}{3^{2}+2}\end{array}[/latex]

The top of the fraction is all set, but the bottom (denominator) has remained untouched. Apply the order of operations to that as well. Begin by evaluating [latex]3^{2}=9[/latex].[latex]\begin{array}{c}\frac{14}{3^{2}+2}\\\\\frac{14}{9+2}\end{array}[/latex]

Now add. [latex]9+2=11[/latex].[latex]\begin{array}{c}\frac{14}{9+2}\\\\\frac{14}{11}\end{array}[/latex]

#### Answer

[latex-display]\frac{5-\left[3+\left(2\cdot\left(-6\right)\right)\right]}{3^{2}+2}=\frac{14}{11}[/latex-display]## The Distributive Property

Parentheses are used to group or combine expressions and terms in mathematics. You may see them used when you are working with formulas, and when you are translating a real situation into a mathematical problem so you can find a quantitative solution.[latex]\begin{array}{c}\,\,\,3\left(2\text{ tacos }+ 1 \text{ drink}\right)\\=3\cdot{2}\text{ tacos }+3\text{ drinks }\\\,\,=6\text{ tacos }+3\text{ drinks }\end{array}[/latex]

The distributive property allows us to explicitly describe a total that is a result of a group of groups. In the case of the combo meals, we have three groups of ( two tacos plus one drink). The following definition describes how to use the distributive property in general terms.

### The Distributive Property of Multiplication

For all real numbers*a, b,*and

*c*, [latex]a(b+c)=ab+ac[/latex]. What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually.

multiply three by (the sum of three and y), then subtract y, then add 9

To multiply three by the sum of three and y, you use the distributive property -

[latex]\begin{array}{c}\,\,\,\,\,\,\,\,\,3\left(3+y\right)-y+9\\\,\,\,\,\,\,\,\,\,=\underbrace{3\cdot{3}}+\underbrace{3\cdot{y}}-y+9\\=9+3y-y+9\end{array}[/latex]

Now you can subtract y from 3y and add 9 to 9.

[latex]\begin{array}{c}9+3y-y+9\\=18+2y\end{array}[/latex]

The next example shows how to use the distributive property when one of the terms involved is negative.

### Example

Simplify [latex]a+2\left(5-a\right)+3\left(a+4\right)[/latex]Answer: This expression has two sets of parentheses with variables locked up in them. We will use the distributive property to remove the parentheses.

[latex]\begin{array}{c}a+2\left(5-a\right)+3\left(a+4\right)\\=a+2\cdot{5}-2\cdot{a}+3\cdot{a}+3\cdot{4}\end{array}[/latex]

Note how we placed the negative sign that was on b in front of the 2 when we applied the distributive property. When you multiply a negative by a positive the result is negative, so [latex]2\cdot{-a}=-2a[/latex]. It is important to be careful with negative signs when you are using the distributive property.

[latex]\begin{array}{c}a+2\cdot{5}-2\cdot{a}+3\cdot{a}+3\cdot{4}\\=a+10-2a+3a+12\\=2a+22\end{array}[/latex]

We combined all the terms we could to get our final result.

#### Answer

[latex-display]a+2\left(5-a\right)+3\left(a+4\right)=2a+22[/latex-display]## Absolute Value

Absolute value expressions are one final method of grouping that you may see. Recall that the absolute value of a quantity is always positive or 0. When you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done.### Example

Simplify [latex]\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}[/latex].Answer: This problem has absolute values, decimals, multiplication, subtraction, and addition in it. Grouping symbols, including absolute value, are handled first. Simplify the numerator, then the denominator. Evaluate [latex]\left|2–6\right|[/latex].

[latex]\begin{array}{c}\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}\\\\\frac{3+\left|-4\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}\end{array}[/latex]

Take the absolute value of [latex]\left|−4\right|[/latex].[latex]\begin{array}{c}\frac{3+\left|-4\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}\\\\\frac{3+4}{2\left|3\cdot1.5\right|-\left(-3\right)}\end{array}[/latex]

Add the numbers in the numerator.[latex]\begin{array}{c}\frac{3+4}{2\left|3\cdot1.5\right|-\left(-3\right)}\\\\\frac{7}{2\left| 3\cdot 1.5 \right|-(-3)}\end{array}[/latex]

Now that the numerator is simplified, turn to the denominator. Evaluate the absolute value expression first. [latex]3 \cdot 1.5 = 4.5[/latex], giving[latex]\begin{array}{c}\frac{7}{2\left|{3\cdot{1.5}}\right|-(-3)}\\\\\frac{7}{2\left|{ 4.5}\right|-(-3)}\end{array}[/latex]

The expression “[latex]2\left|4.5\right|[/latex]” reads “2 times the absolute value of 4.5.” Multiply 2 times 4.5.[latex]\begin{array}{c}\frac{7}{2\left|4.5\right|-\left(-3\right)}\\\\\frac{7}{9-\left(-3\right)}\end{array}[/latex]

Subtract.[latex]\begin{array}{c}\frac{7}{9-\left(-3\right)}\\\\\frac{7}{12}\end{array}[/latex]

#### Answer

[latex-display]\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-3\left(-3\right)}=\frac{7}{12}[/latex-display]## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Screenshot Combo Meal.
**Provided by:**Lumen Learning**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Ex 2: Subtracting Integers (Two Digit Integers).
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Find the Sum and Difference of Three Signed Fractions (Common Denom).
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Ex: Adding Signed Decimals.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Unit 9: Real Numbers, from Developmental Math: An Open Program.
**Provided by:**Monterey Institute of Technology and Education**Located at:**https://www.nroc.org/.**License:**CC BY: Attribution. - Ex: Multiplying Three or More Integers.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Ex: Multiplying Signed Fractions .
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Ex 1: Dividing Signed Fractions.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Ex 2: Combining Like Terms.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Simplify an Expression in the Form: a-b+c*d.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Simplify an Expression in the Form: a*1/b-c/(1/d).
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Simplify an Expression in the Form: (a+b)^2+c*d.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Simplify an Expression in Fraction Form.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Simplify an Expression in Fraction Form with Absolute Values.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution.