# Using Mathematics in Physical Science

by Ron Kurtus (revised 29 November 2013)

Mathematics is used in Physical Science to calculate the measurements of objects and their characteristics, as well as to show the relationship between different functions and properties. Arithmetic, algebra and advanced mathematics may be used.

Arithmetic and algebra are used to establish values and solve simple equations or formulae.

In classical or everyday Physics and Chemistry, normal values are used to solve equations. In Astronomy, distances, sizes and masses are very large. Special nomenclature is required to represent these values. In Atomic Physics and some areas of Chemistry, sizes and masses are small, although quantities may be large.

Arithmetic consists of simple operations with numbers and values. Algebra is used to show relationships before the measured numbers are used for calculations. Higher math is used for complex relationships between properties.

Questions you may have include:

• How is Arithmetic used in Physical Science?
• How is Algebra used?
• How is higher math used?

This lesson will answer those questions. Useful tool: Units Conversion

## Arithmetic conventions

In using Arithmetic, we can add, subtract, multiply and divide numbers. We also use fractions and decimals.

We use the "+" symbol to signify adding two numbers and the "" symbol for subtractions. The "=" symbol means equals and is the result. Thus, 5 + 2 = 7 is 5 plus 2 equals 7, and 6 − 4 = 2 is 6 minus 4 equals 2.

You perform the operations in the order they are listed: 5 + 2 − 3 is done as 5 + 2 = 7 and then 7 − 3 = 4.

### Multiplication and division

We commonly use the "x" symbol to signify multiplication in arithmetic: 2 x 3 = 6. But note that "x" can also be a variable in Algebra and mean something else, so caution must be used. Often people are using * to denote multiplication: 2 * 3 = 6.

In web pages, it is difficult to write the division symbol seen in your textbooks, so "/" is used to denote division: 8 / 4 = 2.

#### Order listed

Multiplication and division operations are done in the order listed. Thus: 6 * 2 / 3 = 4 is performed as 6 * 2 = 12 and 12 / 3 = 4.

#### Use of parentheses

When you combine addition and subtraction with multiplication and division, it can get complex. You still go in the order listed, but parentheses must be used to clump together addition and subtraction terms that go together. Operations within parentheses are done first.

5 * 3 + 7 is different than 5 * (3 + 7). With 5 * 3 + 7, the operations are in order, thus we have 5 * 3 = 15 and then 15 + 7 = 22. With 5 * (3 + 7), you combine those within the parentheses first. (3 + 7) = 10 and then 5 * 10 = 50.

### Fractions and decimals

3 / 5 is 3 divided by 5, but since that does not conveniently work out, we can designate that as a fraction and write it 3/5.

If you do divide it out, you can write the result as the decimal 0.6. Note that it is a good idea to put the 0 in front of the decimal point to avoid confusion. Writing .6 may easily confused with the number 6, if the person doesn't notice the tiny "." symbol.

## Algebra

Algebra uses letters to denote a relationship between characteristics. Usually, they are just abbreviations for the characteristic. For example, energy is denoted by E and velocity by v.

Note that we typically will make the variable in boldface, so that it is easier to distinguish from other items, especially in web pages. Many physics textbooks reserve boldface for vectors.

### Multiplication

Although you can multiply numbers using either x or *, such as 2 x 3 or 5*7, letters are often used to designate something that does not yet have a value assigned. A big problem is in algebra, the letter x is often assign to a variable or unknown value. But also using x for multiplication, you might get 2 x x, which is confusing. Even using 2*x, is cumbersome. So, the algebraic standard is just putting the letters together. 2x is 2 times x. xyz is x times y times z. But this way of writing does not follow with numbers. 23x is not 2 times 3 times x. It is 23 times x.

### Relationships

Newton came up with the relationship between force, mass and acceleration. His equation says that force equals the mass of an object times its acceleration. To avoid writing out this sentence, we use the symbols F for force, m for mass and a for acceleration. Thus, the equation can be written: F = ma.

This allows us to substitute values for two items and get a value for the third. If m = 3 kilograms and a = 2 meters per second per second, F = 3 * 2 = 6 newtons.

Note that you will often see the equation written as F = ma in textbooks. They try to abbreviate using a multiplication symbol by just putting the variables next to each other, assuming you know they are multiplied. In some books, they use a "." between the symbols. To avoid confusion, we will continue to use "*" as multiplication.

### Subscripts

Sometimes you are comparing two or more items with the same characteristic. In such cases, a subscript number or letter can be added to keep things separate. A subscript is a small number or letter after and below the variable.

If you are comparing several forces, you can name one F1, another F2 and so on. Also, we call the force of friction Fr to separate it from another force.

### Squares

A square of a number or variable is it multiplied by itself. For example, 3 squared is 3 times 3 and x squared is x times x.

#### x^2

One way of designating a squared variable is by using ^2. Thus, 3^2 = 9 and x^2 is x squared.

#### x²

A more common way of writing the square of a number is with the superscript 2: 3² = 3 * 3 = 9 and x² = x*x.

#### Raised to a power

You can raise a number to a higher power, but not many common physics equations use that: x4 = x*x*x*x. The number 10 raised to a higher power is a handy way to denote large numbers: 106 is 1 followed by 6 zeros = 1,000,000.

### Square roots

A square root is just the inverse of squaring a number. If 3² = 9, then the square root of 9 is the number that when multiplied by itself equals 9. In other words, 3 is the square root of 9.

Most numbers do not have a simple square root, so most must be determined with a calculator. For example, the square root of 25 is 5 but the square root of 24 is 4.898979...

The symbol for square root used in textbooks does not work well in web pages, so instead we use SQRT to indicate the operation. Thus SQRT(25) is the square root of 25 and SQRT(v/g) is the square root of the result of v divided by g.

Calculus, differential equations and other advanced mathematics are used in advanced Physical Science calculations and equations. They are beyond the scope of our lessons.

One example of where and why advanced mathematics must be used can be seen in the simple gravity equations. F = m*g is the equation for the force of gravity. But that equation is only an approximation for items falling close to Earth. The actual equation varies inversely as the square of the distance apart and is related to the masses of the bodies.

## Summary

Mathematics is used in Physical Science for measurements and to show relationships. Arithmetic consists of simple operations with numbers, and algebra shows relationships--often without numbers. Higher math is used for complex relationships between properties.

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