PTM Arithmetic 1: Decimal System and Calculation

**By: Tao Steven Zheng (郑涛)**

## **Part 1: Exploring the Concepts**

### **Numerals**

The symbols for representing numbers are called **numerals**. Ancient peoples from all over the world came up with different sets of symbols for representing the same numbers.

The ancient Egyptians created a base-ten **additive system**. This means the numbers for the symbols were simply added together. For example, the number 23 is written using two symbols for 10 and three symbols for 1. It does not matter which order the numerals are written: you can write the tens first followed by the ones, or the ones first followed by the tens. In fact, Egyptian hieroglyphs can even be written vertically. The Egyptians used seven distinct symbols for writing numerals:

* 1 is a vertical stroke
* 10 is a hobble for cattle
* 100 is a coil of rope
* 1000 a lotus plant
* 10000 is a bent finger
* 100000 is a tadpole or frog
* 1000000 is a figure of the Egyptian god Hah with his arms raised above his head.

In Figure 2, the number 258458 is written using 32 symbols: two frogs for 200000, five bent fingers for 50000, eight lotuses for 8000, four coils of rope for 400, five hobbles for 50, and eight vertical strokes for 8.

The ancient Babylonians devised a base-sixty **place value system**. They wrote on clay tablets by etching wedged-shaped markings called **cuneiform** with a stylus. For example, the number 172 is written using two symbols representing 60 followed by five symbols representing 10 and two symbols representing 1.

Like the ancient Egyptians, the ancient Greeks adopted an additive system without place value. However, the Greek numerals were not purely base-ten. σλζʹ = 200+30+7

The early Roman numerals was identical to the Greek numerals, except with a different set of symbols. By the time of the Roman empire, the Roman numerals transitioned from a purely additive system to a mixed additive and subtractive system. Thus the number nine, once written as VIIII, became IX.

### **Addition**

Since time immemorial, humans have been counting the things they see, and keeping track of the number of objects counted. Another deer in a herd, another pebble on the ground, another star in the sky, another month gone by, the new item is **added** to the tally. Hence, the act of counting involves a basic principle called **addition**.

When two numbers are added together, the two numbers themselves are called the **addends**; their result is called the **sum**. For example, when 7 sticks and 8 sticks are combined as one group, the group contains 15 sticks.

7
# 8
15

{\displaystyle 7 + 8 = 15 }

The addition of numbers is **commutative**, which means the order in which the numbers are added together does not matter. Given two numbers

a

{\displaystyle a}

 and

b

{\displaystyle b}

, then

a
# b
b
a

{\displaystyle a + b = b + a}

. For example,

7
# 8
15

{\displaystyle 7 + 8 = 15 }

8
# 7
15

{\displaystyle 8 + 7 = 15 }

Therefore,

7
# 8
8
7

{\displaystyle 7 + 8 = 8 + 7 }

.

### **Subtraction**

Subtraction is the opposite of addition. Subtraction means to take away a number from another number. For example, when 2 stones are taken away from 7 stones, there are 5 stones remaining.

7
−
# 2
5

{\displaystyle 7-2=5}

The first number (7) is called the **minuend**, the second number (2) is called the **subtrahend**, and the result is called the **difference**. Unlike addition, subtraction is not commutative! This is because the order of numbers subtracted gives a different result. Given two numbers

a

{\displaystyle a}

 and

b

{\displaystyle b}

such that

a
≠
b

{\displaystyle a \ne b}

, then

a
−
b
≠
b
−
a

{\displaystyle a - b \neq b - a }

.

For example,

7
−
# 2
5

{\displaystyle 7-2=5}

2
−
# 7
−
5

{\displaystyle 2 - 7 = -5 }

Therefore,

7
−
2
≠
2
−
7

{\displaystyle 7 - 2 \neq 2 - 7 }

.
In the example above, the number

−
5

{\displaystyle -5}

 (negative five) is called the **additive inverse** of

5

{\displaystyle 5}

. When **negative numbers** are included to the set of numbers, one can view the subtraction operation as equivalent to the addition of negative quantities.

For example,

10
−
# 7
10
(
−
7
# )
3

{\displaystyle 10 - 7 = 10 + (-7) = 3 }

### **Multiplication**

Multiplication is essentially repeated addition. For example, suppose there are 3 bags each with 4 marbles inside. One can add the 4 marbles three times

4
4
# 4
12

{\displaystyle 4 + 4 + 4 = 12 }

, which is identical to writing

3
×
# 4
12

{\displaystyle 3 \times 4 = 12 }

. Here, the numbers multiplied together are called **factors**, and the result is called the **product**.

Because addition is commutative, multiplication is also commutative. Given two numbers

a

{\displaystyle a}

 and

b

{\displaystyle b}

, then

a
×
# b
b
×
a

{\displaystyle a \times b = b \times a }

.

For example,

3
×
# 4
12

{\displaystyle 3 \times 4 = 12 }

4
×
# 3
12

{\displaystyle 4 \times 3 = 12 }

Therefore,

3
×
# 4
4
×
3

{\displaystyle 3 \times 4 = 4 \times 3 }

.

### **Division**

To divide is to separate a whole into equal parts. Suppose there are 100 coins (each of equal value) to be divided fairly among 5 labourers. Each labourer then receives 20 coins.

100
÷
# 5
20

{\displaystyle 100 \div 5=20 }

Here, the number to be divided (100) is called the **dividend**, the number that divides the dividend (5) is called **divisor**, and the result (20) is called the **quotient**. Division is not commutative, hence,

a
÷
b
≠
b
÷
a

{\displaystyle a \div b \neq b \div a }

.

For example,

10
÷
# 2
5

{\displaystyle 10 \div 2 = 5 }

2
÷
# 10
0.2

{\displaystyle 2 \div 10 = 0.2 }

Therefore,

10
×
2
≠
2
×
10

{\displaystyle 10 \times 2 \neq 2 \times 10 }

.
In the example above, the number

0.2

{\displaystyle 0.2 }

 is called the **reciprocal** or the **multiplicative inverse** of

5

{\displaystyle 5}

. For a given number

a

{\displaystyle a}

, its multiplicative inverse is

a

−
1

{\displaystyle a^{-1} }

(alternatively expressed as

1
a

{\displaystyle \frac{1}{a} }

). Therefore, division can also be viewed as the multiplication of **reciprocals**.

For example,

10
÷
# 2
10
×

1
2

=
5

{\displaystyle 10 \div 2 = 10 \times \frac{1}{2} = 5 }

### **Exponents**

Exponentiation is the repeated multiplication of the same number. For example,

2
5

=
2
×
2
×
2
×
2
×
2

{\displaystyle 2^5 = 2 \times 2 \times 2 \times 2 \times 2 }

where the number

2

{\displaystyle 2}

 is the **base** and

5

{\displaystyle 5}

is the **exponent** (also called the **power**). If the exponent is a **unit fraction**, then it represents a **radical**. For example,

2

1
2

=

2

{\displaystyle 2^\frac{1}{2} = \sqrt{2} }

In general, fractional exponents are used to represent a combination of exponents and radicals:

7

3
5

=

7
3

5

{\displaystyle {7}^\frac{3}{5} = \sqrt[5]{7^3} }

### **Logarithms**

Logarithms is the inverse operation of exponentiation. The exponent

3
4

=
81

{\displaystyle 3^4 = 81}

can be reinterpreted as the logarithm

log
3

⁡

(
81
)

=
4

{\displaystyle \log\_3{(81)} = 4 }

where

3

{\textstyle 3 }

is the base of the logarithm. An alternative way to write logarithms in different bases is to express it as the quotient of two logarithms:

log
⁡

(
81
)

log
⁡

(
3
)

=
4

{\displaystyle \frac{\log{(81)}}{\log{(3)}} = 4 }

More generally speaking,

log
b

⁡

(
N
)

=

log
⁡

(
N
)

log
⁡

(
b
)

{\displaystyle \log\_b{(N)}= \frac{\log{(N)}}{\log{(b)}} }

## **Part 3: Practice Problems**

### **Problem 4**

**Part 1:** Evaluate

64

2
3

{\displaystyle {64}^{\frac{2}{3}} }

in two ways. Show the steps for each method. Which method is simpler?

**Part 2:** Using exponent laws, and without using a calculator, prove

32

4
5

=

64

2
3

{\displaystyle {32}^{\frac{4}{5}} = {64}^{\frac{2}{3}} }

.

**Part 3:** Use logarithms to demonstrate why

64

2
3

=

32

4
5

{\displaystyle {64}^{\frac{2}{3}} = {32}^{\frac{4}{5}} }

.