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Definitions of Basic Terms

Numeration: The process of writing and reading numbers using symbols and words.
Number Base: The number of unique digits (including zero) used to represent numbers in a positional numeral system. Common bases include base-10 (decimal), base-2 (binary), etc.
Numerals: The symbols or characters used to represent numbers. For example, in base-10 the numerals are 0,1,2,3,4,5,6,7,8,9.
Numbers: A mathematical object used to count, measure, and label. They represent quantity, value, or magnitude.

Egyptian Numerals

The ancient Egyptians used hieroglyphic symbols to represent numbers in a base-10 system. Each power of 10 had its own symbol:

Stroke (𓏤) = 1
Hobble/cattle hobble (𓎆) = 10
Coiled rope (𓍢) = 100
Lotus flower (𓆼) = 1,000

Numbers were written by repeating these symbols as many times as needed.

Example: Representing 342

342 = 3 x 100 + 4 x 10 + 2 x 1

So it is written as:

𓍢𓍢𓍢 (3 coiled ropes = 300)

𓎆𓎆𓎆𓎆 (4 hobbles = 40)

𓏤𓏤 (2 strokes = 2)

Combined: 𓍢𓍢𓍢𓎆𓎆𓎆𓎆𓏤𓏤

More Examples

25: 2 x 10 + 5 x 1 = 𓎆𓎆𓏤𓏤𓏤𓏤𓏤

1,000: 𓆼

2,031: 2 x 1,000 + 3 x 10 + 1 x 1 = 𓆼𓆼𓎆𓎆𓎆𓏤

Definitions of Key Terms

Numerals: Symbols used to represent numbers, e.g. 1, 2, 3 or I, II, III.
Numbers: The actual value or quantity being represented. For example, the numeral "5" represents the number five.
Numeration: The system or method of naming and representing numbers, e.g. Egyptian numeration, Roman numeration, Hindu-Arabic numeration.
Number Base: The number of unique digits used in a numeral system. Example: base-10 (decimal) uses digits 0-9, base-2 (binary) uses 0-1.

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Online Chalkboard

Bigdom posts new questions and solutions here regularly for JSS students. Check back for more.

Chalkboard Problem #1

Triangle ABC, Angle B = 90°, Angle A = 50°. Find Angle C.

Solution:

Sum of angles in a triangle = 180°
50° + 90° + C = 180°
140° + C = 180°
C = 180° - 140°
C = 40°

Answer: Angle C = 40°

Number and Numeration (Junior Secondary)

1. Place Value and Face Value

Every digit in a number has two values: place value (the value of where the digit sits) and face value (the digit itself).

Example: In the number 3,456:

The digit 4 is in the hundreds place, so its place value is 400, while its face value is 4.
The digit 5 is in the tens place, place value 50, face value 5.

2. Counting in Base 10

Our number system is base 10. We count from 0 to 9, then add a new place. Numbers can be written in figures (e.g., 247) or in words (two hundred and forty-seven).

Examples: 1,000 = one thousand; 5,678 = five thousand six hundred and seventy-eight.

3. Even and Odd Numbers

An even number ends in 0, 2, 4, 6, or 8. An odd number ends in 1, 3, 5, 7, or 9.

Examples: 2, 4, 6, 8, 10 are even; 1, 3, 5, 7, 9 are odd.

4. Prime Numbers up to 100

A prime number has exactly two factors: 1 and itself. The prime numbers between 1 and 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Note: 1 is not prime.

5. LCM and HCF

HCF (Highest Common Factor) is the largest number that divides two or more numbers exactly. LCM (Lowest Common Multiple) is the smallest number that is a multiple of two or more numbers.

Worked Example: Find LCM and HCF of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6 → HCF = 6
Multiples of 12: 12, 24, 36, 48...
Multiples of 18: 18, 36, 54...
Common multiple: 36 → LCM = 36

Practice these topics to build a strong foundation in mathematics!

History of Numeration

1. Ancient Numeral Systems

Long before modern digits, ancient civilizations developed their own ways to count and record numbers. Each system reflects the culture and needs of its people.

2. Egyptian Numerals

The ancient Egyptians used hieroglyphic symbols to represent numbers. Their system was additive and base-10:

1 – a vertical stroke (like a stick)
10 – a hobble
100 – a coiled rope
1,000 – a lotus flower
10,000 – a bent finger
100,000 – a tadpole or frog
1,000,000 – a god with raised arms

Example: The number 2,345 would be written as two lotus flowers (2,000), three coiled ropes (300), four hobbles (40), and five strokes (5), placed together in that order.

3. Hindu Numerals (Indian System)

Around 500 CE, Indian mathematicians developed the decimal place-value system that we use today. They introduced the concept of zero as both a number and a placeholder. The original Hindu numerals (1–9) evolved from the Brahmi script.

Key feature: A base-10 system with nine digits and a symbol for zero allowed any number to be written simply by position.

Example: 507 means 5 hundreds, 0 tens, 7 ones. The zero in the tens place makes it different from 57 or 5,007.

4. Arabic Numerals (Transmission to the West)

Arab scholars translated Indian mathematical texts and adopted the numeral system, adapting the shapes. The system spread through the Islamic world to North Africa and into Europe. The Italian mathematician Fibonacci (c. 1202 AD) promoted these numerals in his book Liber Abaci, showing their superiority over Roman numerals for calculation.

Example: The number 2,781 in Arabic numerals became the standard across Europe for commerce, science, and everyday use.

5. Roman Numerals

The Romans used letters from their alphabet to denote numbers: I=1, V=5, X=10, L=50, C=100, D=500, M=1,000. They used subtractive notation (e.g., IV=4, IX=9) and additive notation (e.g., VI=6, XI=11).

Examples:

III = 3
IV = 4 (one before five)
IX = 9 (one before ten)
XL = 40 (ten before fifty)
XC = 90 (ten before hundred)
CD = 400 (hundred before five hundred)
CM = 900 (hundred before thousand)
MCMLXXXIV = 1984 (M = 1000, CM = 900, LXXX = 80, IV = 4)

6. Modern Day Numeration

Today, the Hindu-Arabic numeral system is used globally. It is a base-10 (decimal) place-value system. Each digit's value depends on its position: ones, tens, hundreds, thousands, and so on. This system makes arithmetic efficient and universal.

Example: The number 3,402,961 means 3 millions, 4 hundred thousands, 0 ten thousands, 2 thousands, 9 hundreds, 6 tens, 1 one. The zero as placeholder is essential.

Understanding the history of numbers helps us appreciate the power and beauty of mathematics.

Ancient and Modern Numeral Systems

1. Egyptian Hieroglyphic Numerals

The ancient Egyptians used hieroglyphic symbols to represent numbers. Example: 2,345 would be written as 𓆼𓆼𓍢𓍢𓍢𓎆𓎆𓎆𓎆𓏤𓏤𓏤𓏤𓏤. Key symbols:

1: 𓏤 (stroke)
10: 𓎆 (hobble)
100: 𓍢 (coiled rope)
1,000: 𓆼 (lotus flower)
10,000: 𓂭 (bent finger)
100,000: 𓆐 (tadpole)
1,000,000: 𓁨 (god with raised arms)

2. Hindu Numerals (Devanagari Script)

Indian mathematicians developed the decimal place-value system. The original Devanagari digits are:

० १ २ ३ ४ ५ ६ ७ ८ ९

These correspond to 0 through 9.

3. Arabic Numerals (Eastern Arabic-Indic)

Used in many Arabic-speaking countries today:

٠ ١ ٢ ٣ ٤ ٥ ٦ ٧ ٨ ٩

4. Roman Numerals

The Romans used letters: I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1000

Examples: IV=4, IX=9, XL=40, XC=90, CD=400, CM=900, MCMLXXXIV=1984.

5. Modern Hindu-Arabic Numerals

The global standard base-10 system used today:

0 1 2 3 4 5 6 7 8 9

Place value allows any number to be written: e.g., 3,402,961 means 3 millions, 4 hundred thousands, 0 ten thousands, 2 thousands, 9 hundreds, 6 tens, 1 one.

Definitions of Basic Terms

Egyptian Numerals

Example: Representing 342

More Examples

Definitions of Key Terms

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Learning Resources

Mathematics Tutorial Notes

Practice Questions & Answers

Video Lesson Links

Online Chalkboard

Chalkboard Problem #1

Solution:

Answer: Angle C = 40°

Number and Numeration (Junior Secondary)

1. Place Value and Face Value

2. Counting in Base 10

3. Even and Odd Numbers

4. Prime Numbers up to 100

5. LCM and HCF

History of Numeration

1. Ancient Numeral Systems

2. Egyptian Numerals

3. Hindu Numerals (Indian System)

4. Arabic Numerals (Transmission to the West)

5. Roman Numerals

6. Modern Day Numeration

Ancient and Modern Numeral Systems

1. Egyptian Hieroglyphic Numerals

2. Hindu Numerals (Devanagari Script)

3. Arabic Numerals (Eastern Arabic-Indic)

4. Roman Numerals

5. Modern Hindu-Arabic Numerals

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